Skin-Friction and Forced Convection from Rough and Smooth Plates

Since the 1930s, theories of skin-friction drag from plates with rough surfaces have been based by analogy to turbulent flow in pipes with rough interiors. Failure of this analogy at slow velocities has frustrated attempts to create a comprehensive theory. Utilizing the concept of a self-similar roughness which disrupts the boundary layer at all scales, this investigation derives formulas for a rough or smooth plate's skin-friction coefficient and forced convection heat transfer given its characteristic length, root-mean-squared (RMS) height-of-roughness, isotropic spatial period, Reynolds number, and the fluid's Prandtl number. This novel theory was tested with 456 heat transfer and friction measurements in 32 data-sets from one book, six peer-reviewed studies, and the present apparatus. Compared with the present theory, the RMS relative error (RMSRE) values of the 32 data-sets span 0.75% through 8.2%, with only four data-sets exceeding 6%. Prior work formulas have smaller RMSRE on only four of the data-sets.


Introduction
Drag is the pressure opposing fluid flow by an object. Drag consists of "form drag", due to fluid being deflected around the object, and "skin-friction drag" (or "resistance") due to dissipation of the viscous friction stress (shearing stress) generated by flow along the object's surface. Related to skin-friction, "forced convection" is the heat transfer to or from a surface which is induced by fluid flow along that surface. Skinfriction and forced convection are fundamental processes with applications from engineering to geophysics.
Circa 1930, Prandtl [1] and von Kármán [2] developed theories for frictional resistance along (smooth) plates from the results of research on flow through pipes; this is the "pipe-plate analogy".
In 1934, Prandtl and Schlichting [3] developed a theory of skin-friction resistance for rough plates based on their analysis of Nikuradse's [4] measurements of sand glued inside pipes ("sand-roughness"). The conclusion of the (translated) paper states: "The resistance law just derived for rough plates has chiefly validity for a very specific type of roughness, namely a smooth surface to which sand grains have been densely attached and where the Nikuradse pipe results have been taken as the basis. . .
A single roughness parameter (the relative roughness) will in all likelihood no longer answer the purpose in continued investigations of the roughness problem." In 1936, Schlichting [5] investigated the velocity field and skin-friction of water flowing through a closed rectangular channel having one wall replaced in turn by a series of plates, each having an array of identical protrusions attached: spheres, spherical caps (bumps), or cones. The protrusions were positioned on the plates in a hexagonal array which was elongated 15% in the direction of flow.
There was significant pressure drop between the inflow and outflow of the channel, so it was not a case of the isobaric (no pressure drop) flow which occurs along external plates. The similarity of closed channel flow to pipe flow is well known, but does not support or refute treating rough pipe interiors and plates analogously.
In 1954, Hama [6] described three challenges of the pipe-plate analogy: "Now there is no obvious reason why pipe flow and boundary-layer flow should be identical or even similar. First, a pressure gradient is essential for flow through a pipe but not along a plate. Second, pipe flow is confined and perforce uniform, while flow along a plate develops semi-freely and bears no such a priori guarantee of displaying similar velocity profiles at successive sections. Finally, the diameter and roughness size are the only geometrical dimensions of established flow in pipes, whereas at least three linear quantities are necessary to characterize the boundary-layer." Fluid in contact with the plate surface has 0 velocity. The "boundary-layer" is flow near the surface which has velocity different from the bulk flow (or "free-stream") velocity u.
Hama attempted to confirm the rough pipe-plate analogy with measurements of wire screens affixed to smooth plates, but concluded that it was confirmed only in the fully rough regime (defined below).
Testing a machined analogue of sand-roughness, Pimenta, Moffat, and Kays [7] wrote in 1975 that, while the agreement of their data with the Prandtl-Schlichting plate model was "rather good" in the fully rough regime, their apparatus did not have the same behavior as "Nikuradse's sand-grain pipe flows in the transition region." By modeling the wake component of the velocity profile, in 1985 Mills and Hang [8] presented a formula improving the match with Pimenta et al data in the rough regime, but was silent about other flow regimes.
Along with laminar flow, the theory for flow within pipes (and channels) distinguishes three turbulent flow regimes: smooth, (fully) rough, and transitional. Smooth turbulent pipe flow encounters viscous resistance which varies inversely with the fluid velocity per viscosity ratio. Rough pipe flow encounters resistance which varies with the height-of-roughness, while the effect of viscosity is negligible. The transitional regime describes the range of flow rates where both viscosity and roughness affect the resistance (Colebrook [12]).
The Reynolds number (Re) represents the bulk fluid flow rate (far from the plate); Reynolds numbers with subscripts represent other flow rates. The local Reynolds number Re x = x Re/L, where x is the distance from the leading edge of the plate in the direction of flow, and characteristic-length L is the length scale for the physical system. Except where stated otherwise in this investigation, L is plate length in the direction of flow.
Drag in prior works is represented by the local drag coefficient c f or C f , which is a function of relative sand-roughness x/k S , and of the Reynolds number or a subscripted Reynolds number.
Prandtl and Schlichting [3] specified the boundaries between flow regimes using the "sand-roughness Reynolds number" Re k . For plates, they assigned the boundaries between smooth, transitional, and rough regimes at Re k = 7.08 and 70.8. Pimenta et al [7] gave Re k = 65 as the transitional-to-rough boundary.  Figure 1 presents skin-friction coefficient curves, Re k regime boundaries and values, and measurements from a plate having relative sand-roughness L/k S = 1200.
In pipes, the resistance of rough flow is never less than the resistance of smooth turbulent flow occurring in otherwise identical conditions. A tenet of the rough pipe-plate analogy is that this rule also applies to rough, external plates; for example, the "Prandtl-Schlichting c f " curve is never less than the "White (smooth) C σ " curve in Figure 1.
The Pimenta et al measurements are much closer to c f /2 than they are to c f . 1 Pimenta et al [7] and Mills and Hang [8] both referred to skin-friction coefficients as C f /2. All three measurements in Figure 1 are less than the smooth turbulent coefficient C σ . Thus, these measurements can not be in any turbulent flow regime; the remaining alternative is laminar flow. Laminar flow coefficients have a steeper slope than C σ ; yet these measurements are nearly at the constant level predicted by Mills-Hang C f /2 for the rough regime. This is where the pipe-plate analogy fails for roughness. Rough turbulent skin-friction coefficients can take values less than smooth turbulent coefficients for external plates, but not inside pipes.
Nevertheless, research continued based on the pipe-plate analogy. A 2004 survey article by Jiménez [13] did not consider the possibility that the pipe-plate analogy was incorrect, writing: "The theoretical arguments are sound, but the experimental evidence is inconclusive." In 2001, Tachie, Bergstrom, Balachandar, and Ramachandran [15] combined multiple studies to propose a correlation for skin-friction coefficient as a function of the momentum-thickness Reynolds number Re θ : C f (Re θ ) = 4.13 × 10 −2 − 2.68 × 10 −2 log 10 Re θ + 6.528 × 10 −3 log 2 10 Re θ − 5.54 × 10 −4 log 3 10 Re θ ± 7% 150 < Re θ < 15000 (1) As a function of a roughness metric, formula (1) has no predictive value because Re θ must be computed from velocity field measurements along the surface in question.
Circa 2005, Bergstrom, Akinlade, and Tachie [14], performed experiments with sandpapers, woven wire meshes, and perforated sheets attached onto a flat plate, reporting that: where δ * is the displacement thickness and δ is the 99% velocity boundary-layer thickness. As a function of a roughness metric, formula (2) also has no predictive value because both δ * and δ must be computed from velocity field measurements along the surface in question. Fortunately, Bergstrom et al included free-stream velocity in their tables, allowing comparisons of their skin-friction data with the present theory.
Lastly, an additional complication was described by S. F. Hoerner in the discussion section of Hama [6]: "One conclusion from sand experiments has been the expectation that from then on every rough surface should have a constant terminal drag coefficient. As early as 1924, it has been demonstrated (by Hopf and Fromm in Zeitschr. Angew. Math. Mech., 1923:329-339) that certain types of roughness do not show any constant coefficients." The sand-roughness metric's lack of generality, the need for additional roughness parameters, nonconstant terminal drag coefficients, and the failure of the pipe-plate analogy for roughness motivate a fresh theoretical analysis of isobaric flow along a rough plate which is based on a traceable roughness metric.

Roughness metrics
Two established, traceable roughness metrics are the root-mean-squared (RMS) height-of-roughness and the arithmetic-mean height-of-roughness. For an elevation function z(x, y) defined on area A having a convex perimeter, its mean elevation z and RMS height-of-roughness ε are: The arithmetic-mean height-of-roughness is defined in terms of the same mean elevation z: Modeling sand-roughness grains as diameter k S spheres sitting in a pool of depth g glue, the mean elevation z of a cell of area A containing one sphere is: From formula (3), the cell's RMS height-of-roughness ε is: Nikuradse Figure 2 k S /ε versus cell area of sand-roughness Figure 2 shows k S /ε versus the ratio of cell area to the sphere's shadow area, at six glue-levels between 0% and 50% of k S . Table 1 lists k S , grain densities, and conversion factors for Nikuradse's [4] sand coatings, assuming g = 0.5 k S .  [16] fit 5.333 as the RMS to sand-roughness conversion factor k S /ε, and 6.45 as the arithmetic-mean to sand-roughness conversion factor in pipes. k S /ε = 5.333 is a broad minimum of the g = 0.5 k S curve in Figure 2.
The "k S /ε @ g = 0.5 k S " column values in Table 1 ("Nikuradse" in Figure 2) match each other within 2%. The tightest spread on Table 1 data with the arithmetic-mean height-of-roughness exceeds 20%. Thus, sand-roughness correlates an order of magnitude more strongly with RMS height-of-roughness than with arithmetic-mean height-of-roughness.
Flack, Schultz, Barros, and Kim [17] measured skin-friction from grit-blasted surfaces in a duct, writing "The root-mean-square roughness height is shown to be most strongly correlated with the equivalent sandroughness height (k S ) for the grit-blasted surfaces." Arithmetic-mean height-of-roughness will not be considered further by this investigation.
The plate tested by Pimenta et al [7] was composed of 11 layers of densely packed 1.27 mm diameter metal spheres "arranged such that the surface has a regular array of hemispherical roughness elements." Having been brazed together, there was no pool of glue surrounding the spheres. Shrinking the cell to the sphere's shadow, π k 2 S /4, the RMS height-of-roughness of the top half of the 1.27 mm sphere is 0.150 mm. Converting k S = 0.794 mm from Pimenta et al by the 5.333 conversion factor from Afzal et al [16] yields k S /5.333 ≈ 0.149 mm, which matches 0.150 mm within 1%.
Hence, this investigation uses k S /ε = 5.333 as the conversion factor.

Formulas from prior works
In Boundary-layer theory [9], Prandtl and Schlichting gave formulas for fully rough local (c f ) and plate average (c f ) skin-friction coefficient for a rough plate as a function of x/k S and L/k S , respectively.
Mills and Hang [8] gave a formula (9) which is more accurate than formula (7) on the local skin-friction measurements from Pimenta et al [7]. Their local (C f ) and average (C f ) coefficient formulas were: White [10] gave formula (11) for fully rough local skin-friction coefficient: White is also the source of a widely used formula for turbulent skin-friction coefficient of smooth plates: Mills and Hang [8] derived the average formula (10) from the local formula (9) by fitting a curve to the result of the numerical integration in formula (13): The local formulas (7), (9), and (11) each have a singularity where the expression containing the logarithm is 0. The lower limit of integration (L 0 /k S ) must be large enough to avoid this; but the lower limit is not revealed in the prior works. The averaging formula (13) is quite sensitive to the lower limit because the largest value of the local formula occurs there.
For the Mills-Hang formula (9), with lower bound L 0 /k S = 1.6 and initial dx/k S = 0.01, integration of the local C f is within ±0.5% of the average C f in formula (10) over the range 200 < x/k S < 200 × 10 3 .
For the Prandtl-Schlichting formula (7), with lower bound L 0 /k S = 0.5 and initial dx/k S = 0.5, integration of the local c f is within ±0.5% of the average c f in formula (8) over the range 200 < x/k S < 200 × 10 3 .
Churchill [11] compared 8 formulas from various sources with the data from Pimenta et al [7]; it did not find any to be significantly closer to the measurements than the Mills-Hang local formula (9).
Churchill had distinct formulas for computing the average (mean) skin-friction C m from local C f for smooth and rough surfaces, respectively: C These fully rough regime formulas are compared with measurements and the present theory in Section 19.

Rough turbulence
Forced flow along a plate with a rough surface differs in character from forced flow along a smooth surface because roughness disrupts what would otherwise be a viscous sub-layer adjacent to the plate. Lienhard and Lienhard [18] teaches: "Even a small wall roughness can disrupt this thin sublayer, causing a large decrease in the thermal resistance (but also a large increase in the wall shear stress)." Jiménez [13] wrote "In flows with δ/k < 50, the effect of the roughness extends across the boundarylayer, and is also variable. There is little left of the original wall-flow dynamics in these flows, which can perhaps be better described as flows over obstacles." This investigation focuses initially on the case where flow over obstacles dominates the dynamics. It does this by modeling the rough shearing stress as resulting from the interaction of flow with a roughness which disrupts flow at a succession of scales from L converging to 0. While simpler surfaces may also produce rough turbulence, a roughness which disrupts at this succession of scales surely will. Self-similarity, expressed as a recurrence formula, will be used to succinctly describe such a roughness across scales.
The turbulent boundary-layer thickness along a smooth plate increases proportionally to x 4/5 , where x is the distance from its leading edge (Schlichting [9]). Roughness whose envelope height increases linearly with x can disrupt emerging boundary-layers repeatedly along the entire plate.
This approach departs from prior works because their continuous boundary-layer assumption is incompatible with roughness-as-disruption.

Profile roughness
Simpler than surface roughness, roughness profiles can nonetheless be informative.
Let a "profile roughness" be a function z(x) with 0 ≤ x ≤ L. Given a roughness profile z(x), its mean elevation z and RMS height-of-roughness ǫ are computed similarly to surface roughness ε: Let a "self-similar profile roughness" be a profile roughness function z(x) with (integer) branching factor n ≥ 2 such that the RMS height-of-roughness of z(x) over an interval x 0 < x < x n is n times the RMS height-of-roughness of z(x) over each equally divided sub-interval x t < x < x t+1 with 0 ≤ t < n at a succession of scales converging to 0. The x intervals are open; each z(x t ) value contributes to the height-of-roughness of its parent interval, but not to any sub-interval.
A consequence of this definition is that the ratio of the length of an interval per the RMS height-ofroughness of z over that open interval will be invariant through its succession of scales.
Of particular interest are self-similar roughness profiles which are permutations of the linear ramp z(x) = ς x/L with 0 ≤ x ≤ L. Every elevation from 0 to peak height ς occurs exactly once in a ramppermutation.
The only occurrence of x in formula (16) is z(x); hence the RMS height-of-roughness calculation depends only on the z values, not on their relation to x. Thus, the height-of-roughness of any ramp-permutation is identical to the height-of-roughness of the linear ramp: 6. Self-similar ramp permutations A self-similar integer sequence Y (t, w) from integers 0 ≤ t < w = n q allows self-similar behavior to be explored with a finite approximation. Letting t = ⌊w x/L⌋ ≡ floor(w x/L) constructs a roughness profile from a sequence by z(

512)
t = floor(w x / L) Figure 3 Gray-code profile roughness Self-similar recurrence (18) defines the integer Gray-code sequence G(t, w) shown in Figure 3.
otherwise.  Figure 4; it reverses direction at each bifurcation, yielding a wiggliest possible self-similar n = 2 ramp-permutation sequence.

512)
t = floor(w x / L) Figure 5 random reversal bifurcation profile roughness Figure 5 shows a sequence generated by randomly reversing or not at each bifurcation in recurrence (20).
For a given w = 2 q ≥ 4, there are 2 w−1 distinct self-similar n = 2 ramp-permutation sequences, but only 2 distinct ramps and 2 distinct wiggliest sequences.

Friction travel and velocity
In the conversion of bulk flow to rough turbulence, some particles of fluid must move in directions not parallel to the bulk flow. Such movement results from deflections of flow by roughness peaks, pits, ridges, and valleys; the amount of turbulence induced by a roughness should grow as the height-of-roughness increases.
Let "run" be the horizontal axis and "friction" be the vertical axis of a profile roughness such as in Figure 5. For an integer ramp-permutation sequence Y (t, w), the sum of the (dimensionless) lengths of all run segments is w − 1 = 2 q − 1. The sum of the friction segment lengths is: If a particle of fluid traces the ramp-permutation sequence Y (t, w) between t = 0 and t = w − 1, then w − 1 is the run distance it travels, while formula (21) is the friction distance.   Figure 6 shows the friction travel per run travel ratio versus q, the (integer) base-2 logarithm of w. The linear ramp's slope is 0; the Gray-code's is 1/2; the random reversal cases are approximately 1/2; and the wiggliest roughness has slope 2/3.
A wiggliest roughness sequence W (t, w) is an extreme case; it reverses friction direction at each increment of run (t). For each wiggliest roughness sequence with w ≥ 4 there are 2 w−1 − 2 other random reversal roughness sequences. The linear ramp never reverses direction. For each linear ramp sequence there are 2 w−1 − 2 other random reversal sequences. Being outliers, W (t, w) and linear ramps are excluded from further consideration as roughness.
In Figure 6, the friction per run ratios for Gray-code and random reversal sequences are close to: Y (t, w), t, and w are dimensionless. The friction per run ratio (22) needs to be reformulated in terms of the height-of-roughness. Turning to dimensional analysis, the argument to log 2 must be dimensionless, involve ǫ, and be greater than 1, so that the logarithm will be positive. This friction per run ratio must increase with increasing ǫ; thus, ǫ and the logarithm will be in denominators, yielding: Scaling formula (23) by 1/ √ 12 from formula (17) converts formula (23) into the RMS friction travel per run travel ratio: 1 √ 12 Considering the run travel and friction travel with respect to time lets formula (24) also serve as the friction velocity u τ per bulk flow velocity u ratio. 8. Skin-friction from a rough plate Newberry and Savage [19] demonstrated that some self-similar systems which are modeled using continuous power-law probability distributions can be modeled using discrete power-law distributions.
This investigation uses that idea in reverse. The conversion of run velocity into friction velocity by contact with discrete self-similar n = 2 roughness was modeled by formula (24); the u τ /u conversion ratio for a random self-similar roughness will be inferred using a continuous random variable Z > 1 having a Pareto distribution whose frequency of Z is Z −2 : In the generalization of formula (24) to random self-similar roughness in formula (25), the profile height-of-roughness ǫ was replaced by the surface height-of-roughness ε. As with the profile RMS height-ofroughness, the surface RMS height-of-roughness of any permutation of the linear ramp from 0 to ς is ς/ √ 12. Thus, the 1/ √ 3 (= 2/ √ 12) coefficient is the same in formulas (24) and (25) An example of self-similar ramp-permutation surface roughness is shown in Figure 7. A plate surface composed of parallel, edge-to-edge ridges and valleys will disrupt flow which is perpendicular to the ridges, but may not disrupt flow which is parallel to the ridges. To avoid this complication, this investigation considers only surface roughness which is "isotropic": rotating the flow azimuth in the plane of the rough surface does not substantially affect its behavior. 2 The skin-friction coefficient f c is the ratio of the shearing stress τ 2 per the fluid flow's dynamic pressure (kinetic energy density) ρ u 2 /2, where ρ is the fluid density: Both τ 2 and ρ u 2 /2 have units of pressure, kg/(m · s 2 ); hence f c is dimensionless. Multiplication by formula (25) converts bulk flow velocity u into friction velocity u τ , from which τ 2 is derived: Combining equations (26,27) yields the average skin-friction coefficient of an isotropic, self-similar rough surface: Prandtl and Schlichting [3] calculated τ = ρ u τ 2 , not τ 2 = ρ u τ 2 /2. As a result, c f > 2 f ρ and c f > 2 f ρ . Both Pimenta et al [7] and Mills and Hang [8] treat c f /2 as the skin-friction coefficient.
9. Skin-friction from a smooth plate For a given Re ≫ 1 there must be an L/ε ratio so large that an L length plate with self-similar roughness ε has turbulence-inducing behavior midway between that of a rough surface and that of a smooth surface. Let the "roughness Reynolds number" Re ε derive from friction velocity u τ at scale ε: where ν is the fluid kinematic viscosity and Re = L u/ν. The Re strength at which rough plate skin-friction transitions to smooth plate skin-friction will have the same Re ε value at all L/ε ≫ 1. Combining Re ε = 1 with formula (29) relates Re with L/ε at this transition: Relation (30) linking Re with L/ε suggests that the smooth turbulent friction coefficient f σ can be inferred from formulas (28,30). However, there being no roughness on a smooth plate, f ρ formula (28) must be adapted for this application. Scaling f ρ (L/ε) by 3 √ 2, and its argument by 1/e: L/ε can be eliminated from formulas (30,31) using the Lambert W function, the inverse of ϕ exp ϕ. The inverse of a related form, ϑ = ϕ ln ϕ, is ϕ = exp W 0 (ϑ), where W 0 is the principal branch of W.
The (1) is a fixed point of ϕ ln ϕ and its inverse,    [11] compared smooth turbulent skin-friction coefficient formulas from multiple sources with measurements from Smith and Walker [20], and Spalding and Chi [21]. Figure 8 plots average (not local) coefficient formulas from Churchill; the key gives the RMS error of the measurements relative to each formula.
RMS relative error (RMSRE) gauges the fit of measurements g(Re) to a formula f (Re), with each sample having equal weight. The RMS error of g(Re) relative to f (Re) at n points Re j is: Figure 8's key splits RMSRE into bias and scatter. The root-sum-squared of bias and scatter is RMSRE. 0.75% RMSRE versus "present work" formula (32) is less error than any formula evaluated by Churchill.

Spectral roughness
Several prior works [4,5,6,8,12] use the term "uniform roughness" to describe sand-roughness, implying that its height-of-roughness is the same at all scales. This concept of uniform roughness is incompatible with self-similarity; the RMS height-of-roughness of a portion of a self-similar surface must shrink with the succession of scales converging to 0. Sand-roughness could be described as "repeated roughness". However, a roughness composed of parallel rows of 1000 sand grains spanning its length can also be described as having 500 sand grain pairs. An unambiguous method for determining the spatial period is needed.
The discrete Fourier transform converts a series of equally-spaced samples of a (roughness) function into complex coefficients for each of its harmonic sinusoidal components. There is a profound connection between the discrete Fourier transform X j of a roughness sequence Y (t, w) and its RMS height-of-roughness ǫ: elevations have imaginary parts = 0; hence there are w/2 + 1 distinct |X j |; only 0 ≤ j ≤ w/2 needs to be considered in the developments which follow. Constant term X 0 = z, the mean value of Y, is the only X j term not included in the sum for ǫ. Hence, the dominant component of roughness will be the X j (0 < j ≤ w/2) having the largest amplitude. Let the "period index" j P be the index j of this X j having the largest amplitude |X j |.
When one |X j | is dominant, j P is well-defined and the roughness profile's spatial period is L P = L/j P . Gray and random spectra Figure 10 Gray and random eighths Figure 9 shows the |X j |/w amplitude spectrum of the Gray-code roughness profile from Figure 3; also the mean Fourier spectrum amplitudes from 187 instances of 128-point random reversal profiles (w = 128). For both spectra, X 1 has the largest amplitude; thus j P = 1, indicating that neither spectrum is from repeated roughness. Figure 10 shows the spectrum of eight concatenated repetitions of a Gray-code sequence, and the mean Fourier spectrum amplitudes from 187 instances of eight concatenated random reversal sequences. The period index j P of the Gray-code eighths is 8, as expected; but the random reversal sequences have j P = 6 because their amplitudes are not correlated between the random eighths.
Use of the discrete Fourier transform seems appropriate for detecting and quantifying the spatial period of profile roughness. The next section generalizes this spatial period metric to isotropic surface roughness. 11. Periodic roughness Let a "periodic roughness" be a flat surface tiled with many isotropic, uniformly sized patches, all sharing the same mean elevation z and the same RMS height-of-roughness ε. The mean elevation and RMS heightof-roughness of the surface as a whole will also be z and ε, respectively.
Let S s,t be a w × w matrix of mean elevations from a w × w square grid of an L w × L w region of a rough surface. Its 2-dimensional discrete spatial Fourier transform is: Let the 2-dimensional period index j P = j 2 + k 2 > 0, where 0 ≤ j ≤ w/2 and 0 ≤ k ≤ w/2 are the indexes of the coefficient X j,k having the largest amplitude, excluding X 0,0 . The 2-dimensional spatial period L P = L w /j P .  Figure 11 bi-level plate 50% Figure 12 bi-level plate 25% Figure 11 shows the j P values for a square equal-area bi-level surface (regular array of square posts on a flat plate) computing X j,k from a 64 × 64 interpolated sampling of that surface with azimuth from 0 • through 45 • , and which is scaled between 9 and 13 cells on a side. At each scale, j P varies within a ±1 range versus the azimuth. Figure 12 shows a 25% high, 75% low, bi-level surface; some of its j P traces have peaks outside of the ±1 range. This suggests a quantitative criterion for surface roughness isotropy: A surface roughness is isotropic if j P varies no more than ±2 through its full rotation. More specifically, using w × w (w ≥ 64) samplings of roughness at nine scales over a 2:1 range such that most of the calculated j P values satisfy 8 ≤ j P ≤ 16 at each scale, do 56 sampling trials with randomized offset and randomized azimuth rotation per scale. A roughness is considered isotropic if no more than 5 of the 504(= 56 × 9) trials have j P varying more than ±w/32 from its scale.
Using this criterion, square post arrays having an upper area fraction between 27% and 76% are isotropic. Square arrays of circular columns having an upper area fraction between 24% and 78% are isotropic. Hexagonal arrays of circular columns having an upper area fraction between 6% and 75% are isotropic.
Flipping the elevations, hexagonal arrays of circular depressions in a flat plate having an upper area fraction between 25% and 94% are isotropic. Hexagonal arrays of square columns having an upper area fraction between 4% and 49% are isotropic.
Plates with 15% elongated hexagonal arrays of cone or bump protrusions described in Schlichting [5] are also isotropic; with rotation, L P varies between the periods of the elongated axis and its perpendicular.
This isotropy test is not equivalent to the visual appearance of isotropy. Square post arrays having upper area fractions of 20% fail the isotropy test, while those with 30% pass. Plates from Schlichting [5] having elongated hexagonal arrays of cones are not visually isotropic, yet pass the isotropy test.
A plate surface is isotropic, periodic roughness when its j P (≡ L/L P ) ≫ 1.

Onset of rough turbulence
For an isotropic, periodic roughness with 0 < ε < L P ≪ L, there must be some value Re λ > 0 such that when 0 < Re < Re λ , the fluid flow exhibits laminar or smooth turbulent behavior along the entire plate.
The boundary-layer is thinnest at the leading edge. Because the roughness is periodic, any disruption will start within the leading band (0 < x < L P ) of roughness. In this investigation, a boundary-layer is considered disrupted when ε > δ 2 (L P ), where δ 2 (x) is the boundary-layer momentum thickness at x. δ 2 (x) is the thickness of bulk flow having the same momentum flow rate as the plate's boundary-layer at x. Momentum thickness δ 2 is not a directly measurable quantity. Schlichting [9] gives the momentum thickness of laminar and smooth turbulent boundary-layers as formulas (36) and (37), respectively: Laminar δ 2λ formula (36) is from the Blasius boundary-layer model, which is well established. Smooth turbulent δ 2σ formula (37) is less certain.
Momentum thickness δ 2 (x) is a local property of the fluid flow. In order to work locally with roughness Reynolds number Re ε formula (29), change L → x and Re → Re x ; then solve Re ε = 1 for x/ε: Re x grows proportionally to x; hence, the smooth turbulence momentum thickness δ 2 (x) should be proportional to the product of x and formula (25) u τ /u, the friction travel per run travel ratio: The coefficient seems to be 1/3 3 ≡ 1/27 ≈ 0.0370: The occurrence of 3 3 may be related to the equivalence of ϕ ϕ and exp(ϕ ln ϕ).
Thus, δ 2σ (L P ) is a reasonable approximation for δ 2 (L P ) in the leading band of roughness. Figure 14 shows the (theoretical) momentum thickness of laminar and smooth turbulent flows along a 1 m long plate versus Re. The intersecting laminar δ 2λ and smooth turbulent δ 2 curves partition the graph into four regions marked I, II, III, and IV.
When the point at coordinates [Re, ε] is in region I, the roughness is sufficient to disrupt both laminar and smooth turbulent flow; hence, leading band fluid flow will be rough turbulent.
When [Re, ε] is in regions II or III, the roughness is not high enough to significantly disrupt laminar flow; hence, leading band fluid flow will be laminar, possibly transitioning to smooth turbulent.
When [Re, ε] is in region IV, the roughness would be sufficient to disrupt laminar flow, but not large enough to disrupt smooth turbulent flow; hence, leading band fluid flow would be smooth turbulent.
With δ 2λ = ε and x = L P , solve δ 2λ formula (36) for the laminar upper-bound Re λ : With x = L P and δ 2 (x) = ε in δ 2 formula (40): The inverse of ϕ = W 0 (ϑ) is ϑ = ϕ exp ϕ. Solving for the smooth turbulent upper-bound Re σ = Re: Equating Re λ and Re σ yields their intercept (found numerically): Region IV requires L/ε > 20 L P /ε ≈ 3886. Machining precision must be much finer than this so that the roughness of each patch is larger than residual elevation errors. Region IV smooth turbulent flow will be rare; most isotropic, periodic roughness will transition directly from laminar to rough turbulent flow.

Plateau roughness
Disjoint flow mode regions can form along plates whose roughness peaks are all co-planar (at the same elevation) plateaus. With Re > max(Re σ , Re λ ) producing rough turbulence in the leading band of the plate, smooth turbulent flow occurs downstream from where δ 2 (x) is large enough to bridge the gaps. The drag from the downstream portion of the surface will be proportional to f σ , not the constant f ρ of rough turbulent flow. 3 Informally, a "plateau roughness" is an isotropic, periodic roughness with most of its area at its peak elevation. Of particular interest are plateau roughnesses where each cell contains a contiguous plateau area whose boundary has a convex perimeter within the cell. This will either be an array of "islands" whose tops are all co-planar, or an array of "wells" dropping below an otherwise flat plane.
Consider a smooth flat plate etched with a square grid of grooves subjected to a Re > max(Re σ , Re λ ) flow. When the boundary-layer is disrupted by a groove perpendicular to the flow, the smooth turbulent boundary-layer restarts at the leading edge of the next island. At the scale of the roughness period L P , the momentum thickness of the boundary-layer grows from 0 to nearly the L-scale δ 2 (x) value (depending on the size of the island). If it grows to exceed ε, then the rest of the plate (to its trailing edge) will have a smooth turbulent skin-friction coefficient proportional to f σ .
Along isotropic roughness, the growth of δ 2 depends on plateau size, but not on orientation. In natural convection from an upward-facing horizontal plate [22,23], the (isotropic) characteristic-length metric L * = A * /p * is the convex area A * per its perimeter length p * ratio. For a regular polygon or circle, L * = r/2, where r is the minimum radius of the regular polygon or circle.
Multiplying both sides of the smooth turbulent momentum thickness δ 2 equation (40) by 3 3 Re/[ √ 3 L] allows Re x to be isolated using the Lambert W 0 function identity ϕ/W 0 (ϕ) = exp W 0 (ϕ): Let δ 2 = ε 2 /L * in order to find the island Re x threshold Re I . The Re strength needed to produce smooth turbulence grows with with L/L P . Letting Re = [L/L P ] 3 in equation (46), then taking the logarithm of both sides: The inverse of ϕ = W 0 (ϑ) is ϑ = ϕ exp ϕ. Solving formula (47) for Re I = Re x : With large enough gaps, the islands are too short to support smooth turbulence bridging the gaps, leading to the inequality in formula (48). Formula (48) is tested against two square-grooved plates in Section 23.
Laying a perforated sheet on a flat plate turns its holes into wells. L * = A * /p * is the hole area A * per its perimeter length p * ratio. Flowing fluid must rise after diving into a well; hence, there are twice as many roughness interactions per cell. Instead of Re = [L/L P ] 3 in formula (47), let: leading to the wells Re x threshold Re W : With large enough wells, the flats between them are too narrow to support smooth turbulence bridging the wells, leading to the inequality in formula (50). Formula (50) is tested against perforated sheets in Section 22.
When Ω > 1/2 and Re > max(Re σ , Re λ ), flow along the entire surface will be rough turbulent. Thus it will have a constant terminal drag coefficient.
Sand-roughness and the sphere roughness of Pimenta et al have Ω ≈ 0.877 > 1/2. Therefore, they are not plateau roughness, and will have constant terminal drag coefficients.

Local skin-friction
Conversions between local and average skin-friction are needed in order to compare prior with present work.
The ratio of average to local skin-friction f σ /f σ > 1 of a continuous boundary-layer is calculated from f σ by formula (54). The singularity (division by 0) at the leading edge of the plate is avoided by using Re 0 > 0 as the integration lower bound; Re 0 must be chosen such that f σ (Re 0 ) ≪ 1.
The local skin-friction coefficient is not well-defined for self-similar roughness because its L/ε is constant. Periodic roughness has varying L/ε; conveniently, it also provides L P /ε as the lower-bound of integration.
A crucial difference between periodic roughness and smooth surfaces is that periodic roughness disrupts the boundary-layer repeatedly. Thus, the local skin-friction coefficients being averaged are independent. Instead of f ρ /f ρ accruing linearly, it should go as the square: Applying formula (55) to the Mills-Hang local formula (9) yields the average skin-friction coefficient: where C f and C f are from equations (9) and (10), respectively. At L/ε = 4000 (L/k S = 750), formula (56) has values in excess of 20% larger than the Mills-Hang C f formula (10) derived using averaging formula (13). The average skin-friction coefficient was of minor importance in Mills and Hang [8]; the paper did not make comparisons with measurements of average skin-friction.
For disrupted boundary-layers, the transform for local friction f ρ given average friction f ρ is: Applying formula (57) to f ρ formula (28) for rough surfaces yields the local skin-friction coefficient: The local skin-friction coefficient for smooth turbulent flow f σ can be derived from f σ equation (32) and equations (59) provided that Re x ≥ Re 0 ≥ √ 3 e and Re x ≫ √ 3 e: Figure 15 plots local smooth turbulent skin-friction coefficient formulas and measurements collected in Churchill [11]. The measurements have 1.9% RMSRE versus "present work" formula (60).
When calculating RMSRE, the error due to variation at a single point is larger than the error when that variation is split among multiple points. Thus, local measurements tend to have larger RMSRE than average measurements do. In Figure 15, the primary error contributor is the value at Re x = 10 7 .  Heat transfer by rough turbulence will grow with the amount of turbulence induced, f ρ Re. It will also grow with P r 1/3 . The fluid's Prandtl number (P r) is the kinetic viscosity per thermal diffusivity ratio. Fluids with large P r transport heat in the fluid flow; conduction is dominant in fluids with small P r. The 1/3 exponent for P r may relate to it being a property of the fluid in three dimensions. Heat transfer from the trailing part of the plate is reduced due to fluid heating by the leading part; hence, convection will be scaled by 1/2. Expanding f ρ from equation (28) yields formula (61) for rough turbulent average forced convection: Nu ρ = f ρ Re P r 1/3 2 = Re P r 1/3 6 ln 2 (L/ε) The present analysis for self-similar f ρ did not involve boundary-layers; hence, it avoids the P r > 0.6 restriction affecting formula (62). Formula (61) is compared with measurements in Section 23.

Forced convection from an isothermal smooth plate
For smooth turbulent convection, Lienhard [24] recommends composing the Gnielinski formula (62) with the White smooth formula (12) for C σ .
Lienhard states that Nu formula (63) has similar accuracy for gases: Nu = 0.037 Re 4/5 P r 0.6 0.6 ≤ P r < 2 Forced convection is similar to natural convection from a vertical plate; in both, fluid flows parallel to the plate's characteristic-length axis, and is uniform across the plate's other axis.
Jaffer [25] finds that stationary fluid conducts heat from the plate with an effective Nusselt number Nu 0 = 2 4 /[π 2 4 √ 2] ≈ 1.363; it is a coefficient factor of both the static and flow-induced heat transfer terms. Flow-induced heat transfer grows with Nu 0 Re f σ . Because this heat traverses boundary-layers, the P r dependence is more complicated than P r 1/3 . In Jaffer [25], the vertical-plate natural convection dependence is 3 P r/Ξ(P r), where Ξ(P r) is defined using ℓ p -norm definition (65) with p = 1/3:  Figure 16 smooth plate Nu σ dependence on P r In Figure 16, 3 P r/Ξ(P r) is asymptotically P r 1/3 for large P r and 3 √ 2 P r 2/3 for small P r. At small P r, conduction does not amplify forced convection as it does for natural convection; the P r exponent should be 1. An additional factor using the ℓ 3 -norm accomplishes this. Formula (67) is asymptotically 3 √ 2 P r for P r ≪ 0.5. The "0.7 P r 0.6 " trace shows that formula (67) has a slope close to formulas (63, 64) for gases. 3 P r Ξ(P r) The slope of formula (62) Nu(C σ ) decreases with increasing P r; for large P r, Nu ∝ √ C σ . Recall from equations (26,27) that f c ∝ u 2 τ ; hence f c ∝ u τ . This indicates that transport through the boundary-layer limits convection at large P r. In order to reduce the asymptotic dependence from f σ to f σ , the convection formula will include a factor which takes the square-root of an expression gating f σ by P r: The scaling for upstream heating was 1/2 in Nu ρ formula (61) for disruptive roughness; the smooth turbulent boundary-layer reduces this interaction; 1/3 ≈ 0.577 appears correct in the smooth case.
Lienhard [24] compares the Gnielinski-White convection formula with local measurements from studies of fluids with 0.7 ≤ P r ≤ 257 spanning 4000 < Re x < 4.3 × 10 6 . The smallest turbulent Re x was ≈ 10 5 . Formula (63) is more accurate than Gnielinski-White for turbulent air (P r ≈ 0.71) at Re x < 10 5 . Figures 17 and 18 show Nu σ versus P r and Re, respectively. The "present work" traces are formula (69). The "averaged" traces use formula (70) to numerically average the composition of the Gnielinski formula (62) with the White formula (12).

Periodic smoothness and rough laminar flow
Flow along a flat, single-crystal silicon wafer should exhibit smooth formula (72) behavior. At atomic scale, however, it is an isotropic, periodic roughness with L p = 543 nm and ε = 31.2 nm. Is it rough or smooth? The amount of laminar flow displaced by a periodic roughness grows with both ε and L P . Dimensional analysis suggests that this displacement is significant when L/ √ ε L P < Re σ . Plates with L/ √ ε L P > Re σ can be considered "smooth", behaving as formula (72) when Re < Re λ . Table 2 compares parameters of several surface geometries. The "silicon wafer" row has L/ √ ε L P > Re σ , indicating that it is smooth.  Table 2 dimensionless parameters Now consider L/ √ ε L P < Re σ . Tuck and Kouzoubov [26] finds that a slow laminar flow over a periodic roughness ". . . represents a shift of the apparent plane boundary toward the flow domain." At small Re, the rough laminar flow from the apparent boundary plane outward is the same as smooth laminar flow.
Between the surface and its apparent boundary plane, however, shearing stress periodically (L P ) exceeds that of a smooth surface, becoming more likely to spawn vortexes as Re increases, asymptotically approaching smooth turbulent flow. This "rough laminar" convection is a case of heat transfer "processes which vary uniformly between . . . limiting cases" (Churchill and Usagi [27]). Jaffer [25] models the combination of such processes using the ℓ p -norm formula (65) with p > 1 when the processes are competitive.
If the laminar and turbulent flows were perpendicular, their convection processes would combine as root-sum-squared, the ℓ 2 -norm; but these flows are aligned. Independent processes combine as the ℓ 1 -norm (addition); but these processes are competitive. An intermediate strength is needed. The geometric mean of 1 and 2 is √ 2; hence, the ℓ f λ , Nu λ , and Nu λ derive from the Blasius laminar model in Schlichting [9]. Section 24 tests formula (74).
Isotropic, periodic roughness with Ω < 1/2 is plateau roughness, shedding either smooth or rough turbulence, or a combination, when Re > Re λ . Table 3 proposes the behaviors of plateau islands and plateau wells when Re > Re λ . The uppermost flow mode satisfying the inequality to its left takes precedence. The Re x conditions split the plate at distance x from the leading edge into regions operating in different modes. The islands threshold Re I is formula (48); the wells threshold Re W is formula (50  Table 3

turbulent flow modes for plateau roughness
The "Re x > Re I " condition for islands is smooth turbulence with characteristic-length L P . The island's area is augmented by 1/2 of the non-plateau area combined with 2 ε [4 L * ] using the ℓ 2 -norm: Formulas (78) and (79) are the skin-friction and convection predicted for the entire active surface. The ℓ p -norm exponent p = −16 if f ρ > df I /dRe at Re = Re I ; and p = +32 otherwise. Figure 21 shows that Nu and f c are closely related, particularly at Re I . Figure 21 indicates that p = +32 surfaces have constant terminal drag coefficients, while p = −16 surfaces do not.  In plateau wells "blend" mode, the perforated sheet's plateau area is shedding smooth turbulence while its wells are in rough turbulence. The effective friction coefficient is the area-proportional blend: Where Re x > Re W along a perforated sheet, the friction is smooth turbulent, but with additional area 2 π ε [4 L * ] from the cylindrical walls. Although the walls and plateau are perpendicular, the same flow brushes by both. A strength between ℓ 2 and ℓ √ 2 is needed; their geometric mean is p = 4 √ 8 ≈ 1.682:  Figure 22 average friction coefficient of sand-roughness Figure 22 compares fully rough regime average skin-friction formulas. To the right of each "∈ ±" is the maximum discrepancy from "f ρ (present work)" equation (28) over the Mills-Hang range 750 < L/k S < 2750, which is 4000 < L/ε < 14666.
Starting from the bottom of the Figure 22 key, "0.5 C f " is 1/2 of the Mills-Hang average formula (10). The two "0.5 C m (C f )" traces are 1/2 of the Churchill smooth and rough averaging formulas (14) and (15) applied to the Mills-Hang local formula (9). "0.5 c f " is 1/2 of the Prandtl-Schlichting average formula (8).
The points labeled "Pimenta et al" in Figure 23 show the local drag coefficient measurements versus x/k S for the sphere-roughened plate at three rates of flow, Re k = 41.6, 68.5, and 103. Table 4 gives the RMS error of this set of 19 measurements relative to each of the rough regime formulas: Formula (38) calculates x/ε from the Re x and Re ε = Re k /5.333 values supplied by Pimenta et al [7]. Figure 24 plots local f ρ (x/ε) versus Re x (grouped by Re k ) for the sphere-roughened plate. Compared with the present theory, these measurements have an RMSRE of 4.5%, −2.5% bias, and 3.7% scatter. The bias and scatter differ from Table 4    Having L P /ε < 194, Re λ formula (42) predicts that fluid flow along the entire sphere-roughened plate transitions directly from laminar to rough turbulence at Re λ ≈ 24640; but the smallest reported Re x was 3.8 × 10 5 .

Smooth plate and sandpaper
Bergstrom et al [14] has skin-friction coefficient measurements of sandpapers, woven wire meshes, and perforated sheets attached to a smooth plate, and also the 1.67 m × 1.16 m smooth plate alone. Skin-friction measurements were derived from Pitot probe measurements of air velocity at locations which were 1.3 m downwind from the leading edge of the plate. Bergstrom et al [14] estimated 5% as the combined measurement uncertainty of the smooth surface friction coefficient, and 9% for the rough surfaces.
The measurement tables in [14] include a column for free-stream velocity, U e . In order to compute the Reynolds number Re = U e L/ν from U e , the kinematic viscosity ν = 16 × 10 −6 m 2 /s was calculated for air at 20 • C and the mean atmospheric pressure at the University of Saskatchewan, 95 kPa. Microscopic examination of coarse grades of sandpaper reveals glued mounds of grits and canyons having depths which are several times the mean grit diameter. Sandpaper grit mean diameter is standardized, but not the height-of-roughness of the mounds; it can vary by manufacturer and lot. The horizontal traces in Figure 25 show that skin-friction coefficients which are independent of Re x , as in the present theory, can be within the 9% estimated measurement uncertainty of the data.
The RMS height-of-roughness ε of sandpaper is much larger than ε of sand-roughness with the same mean grit diameter. For example, 40 grit sandpaper has a skin-friction coefficient consistent with ε ≈ 503 µm, while k S = 425 µm sand-roughness would have ε = k S /5.333 ≈ 80 µm.  Figure 26. A 0.36 mm wire diameter has conventional openness [s − d] 2 /s 2 ≈ 44% and matches the WMS data and the WMM trace and data.
There are periodic gaps between the wires and the plate; so the mesh-plate combination is not strictly a roughness. With the gaps filled, the RMS height-of-roughness ε would be: The periodic gaps between wires and the plate increase the flow's shearing stress. Scaling ε by the square root of the filled-gap side area per side area ratio is an increase of about 26% for these meshes: The "unscaled 1.04 mm, 3.68 mm" trace in Figure 26 shows the predicted WML friction without this scaling.  Figure 26 local f ρ versus Re x of woven wire mesh Using (scaled) ε ′ , the WML and WMM measurements match the present theory well within the ±9% estimated measurement uncertainty. The WMS measurements do not match unless a hypothesized single digit misprint in [14] is corrected, changing the wire diameter from 0.036 mm to 0.056 mm.
Taken together, the three (corrected) wire meshes have 3.3% RMSRE versus the present theory.  Table 6 perforated sheet openness Table 6 checks the openness of the perforated sheets from Bergstrom et al [14]. It finds that the holes were hexagonally arrayed. However, the PS sheet's calculated openness is 1/2 of the paper's 22%. There are two single digit changes, either of which results in hexagonal openness near 22%: hole diameter d = 1.7 mm or center spacing s = 2.4 mm.
North American suppliers of perforated sheet metal generally specify hole diameter and center spacing in terms of 1/64 of an inch.  Table 7 perforated sheet parameters Table 7 shows the dimensions and metrics of the perforated sheets when laid on the flat plate. For PL, [4 L * ] 2 /L 2 P = d 2 /s 2 ≈ 0.507 > 1/2; its flow will be rough turbulent. The "f ρ (2898) = 0.0052" trace in Figure 27 shows the predicted local skin-friction coefficient's close proximity to the PL measurements. The "× 2.00 mm, 2.81 mm, 0.90 mm" trace with transition at "PL×" shows the behavior predicted if d 2 /s 2 < 1/2. PM and PS have d 2 /s 2 < 1/2. As Re x grows to exceed Re W , the local drag coefficient switches from blend formula (81) to smooth formula (82). The Re W formula (50) transitions are marked by arrows.   Figure 27 local f ρ versus Re x of perforated sheet PL and PM measurements match the present theory within the ±9% measurement uncertainty. The PS measurements do not match unless a hypothesized single digit misprint in [14] is corrected, changing the PS hole spacing from 3.4 mm to 2.4 mm. The "PS×" trace shows the behavior predicted of the original PS. Taken together, the three (corrected) perforated plates have 4.2% RMSRE versus the present theory.
The non-horizontal traces for PM and PS show that they lack constant terminal drag coefficients.

Forced convection measurements
Both the Pimenta et al [7] and Bergstrom et al [14] measurements are restricted to flow rate ranges of less than 3:1. For a novel theory to be persuasive, confirmations over a wider range of flow rates are needed. The present apparatus combined an open intake wind-tunnel, digital phase-locked loop (PLL) fan control, and a heated bi-level aluminum plate having ε = 3.0 mm. This combination measured average convection in air at 2300 < Re < 92000, a 40:1 range. Section 29 describes the apparatus and measurement methodology.
The L/ε ≈ 102 ratio corresponds to a relative sand-roughness of L/k S ≈ 19.
With such a rough plate, the two edges of the bi-level plate roughness which are parallel to the fluid flow also contribute to forced convection, increasing the effective width of the bi-level plate by 4 ε (4%). But the flow near the corner edge is also disrupted. Turning to dimensional analysis, the effective width of the plate increases with the weakly cooperating actual width L W and ε, leading to L W , ε √ 1/2 , about 5.4%. Applying average convection formula (61) to the bi-level plate geometry, with the 5.4% increase, yields: Nu ρ (Re) = 0.00823 Re P r 1/3 (85)  Figure 28, the points labeled "measured" are the Nu measurements with the plate averaging 11 K warmer than the ambient air. The "ℓ 4 mixed" trace shows the ℓ 4 -norm of expected natural and forced convection. At Re < 5000, the natural convection component dominates the mixture.
The "rough turbulent L" trace in Figure 28 is equation (85); the "measured" points match it with 2.2% RMSRE in the range 5000 < Re < 50000.
At Re > 60000, the "smooth turbulent L P " trace is formula (77); its 4/5 slope shows that convection is from a smooth turbulent flow. Its height above the "smooth turbulent L" trace shows that it is operating with a shorter characteristic-length than "smooth turbulent L".
The Nu = 404 value at Re = 58145 is repeatably about 4% lower than expected for an abrupt rough-tosmooth turbulent transition at Re I , and is outside its expected uncertainty bounds (Table 13). Combining the convection formulas (85, 77) using formula (79) and p = −16: The inset graph from 50000 ≤ Re ≤ 90000 shows both abrupt and gradual rough-to-smooth turbulent transitions. The gradual transition is in better agreement with measurement than the abrupt transition.
After collecting convection data, the ε = 3 mm aluminum plate was milled to reduce its roughness to ε = 1.04 mm, resulting in a rough-to-smooth turbulent transition Re I = 6178. Figure 29 shows convection from the plate. Convection measurements at 10 4 < Re < 10 5 match Nu ρ (Re I ) + Nu I (Re) − Nu I (Re I ) with 1.3% RMSRE.
While the "smooth turbulent L P " curve bends downward at Re I in Figure 28, the "mixed" and "forced" curves in Figure 29 bend upward at Re I .
Negative p-norms favor the lesser values, which is why p = −16 was used in formula (86). The upward curvature at Re I in Figure 29 requires a positive p. The ℓ 16 -norm drives some Nu values out of the expected uncertainty bounds. The ℓ 32 -norm fits well with the data in Figure 29: The "ℓ 2 mixed" trace shows the ℓ 2 -norm of expected natural and forced convection with rough turbulence at Re < Re I , and smooth turbulence at Re > Re I ; the "forced; ℓ 32 blend" trace shows a small part of the ℓ 32 -norm blend near Re = 1.13 × 10 5 . The inset graphs expand 5000 < Re < 9000 and 80000 < Re < 10 5 , showing the lower and upper blends, respectively. The present apparatus can not produce the Re > 10 5 flow rates needed to verify the upper blend behavior. Re λ formula (42) predicts that the flow along the entire 3 mm bi-level plate transitions from laminar to rough turbulence at Re λ ≈ 28.8, which is too slow to test in this apparatus. Formula (42) predicts Re λ = 1473 for the 1 mm bi-level plate. The plate was found to have convection consistent with rough turbulence at Re = 2415, which is less than twice its predicted Re λ transition.

Duck tape
The bi-level test plate of the present apparatus has four sides perpendicular to the test surface, each a wedge of extruded polystyrene foam (XPS) insulation filling a 2.7 cm 45 • chamfer in the metal slab. In order to isolate the convective heat flow of the test surface from that of the sides, the estimated side convection, about 10% of total, is deducted from the measured heat flow (see Section 29 for details). Figure 31 rolls of duck tape The surface of the XPS foam board in Figure 30 was not smooth and not an isotropic, periodic roughness. Without a theoretical basis for computing its convection, the accuracy of measurements of the surface-undertest would have been limited.

Figure 30 XPS foam board
The duck tape shown in Figure 31 is a thread mesh (cotton duck cloth) in adhesive on a thin plastic sheet. The vertical periods are longer than the horizontal periods. Using woven wire mesh formula (83) with the geometric mean of the spatial periods as s ≡ L P , the thread diameter plus the sheet thickness as d, and L = 0.305 m yields Tables 8 and 9; Table 9 is identical to the "duck tape" rows in Table 2 1.32 × 10 2 8.82 × 10 3 1.08 × 10 3 < 6.70 × 10 3 ≪ 2.59 × 10 5 duck tape (shiny) 1.54 × 10 2 7.08 × 10 3 1.04 × 10 3 < 2.49 × 10 3 ≪ 1.43 × 10 5 Table 9 duck tape parameters An L = 0.305 m strip of either tape fails to generate rough turbulence because the apparatus's largest Re ≈ 92000 < Re λ . Both tapes have L/ √ ε L P < Re σ , for which the present theory predicts rough laminar formula (74) convection.
With duck tape applied to the foam faces, the measurements presented in Section 23 are consistent with the side convection modeled by rough laminar formula (74). The other curves in Figure 20 are substantially less than formula (74); they would not account for enough heat transfer to keep the surface-under-test measurements within the expected uncertainty bounds presented in Section 29.
Representing only about 10% of the forced convective heat flow from the plate assembly, this evidence is not conclusive for rough laminar formula (74); but it does support the assertion that (forced) laminar convection from a rough surface is larger than either laminar or turbulent convection from a smooth surface of the same size.

Discussion
Rather than trying to tease rough turbulence from a nearly smooth surface, this investigation started its analysis with roughness thicker than smooth plate viscous sub-layers. Recapitulating the development: Self-similar roughness repeatedly disrupts emerging boundary-layers along a plate. Self-similar roughness has a constant characteristic-length per RMS height-of-roughness ratio L/ε ≫ 1 at a succession of scales converging to 0. This leads to friction velocity u τ = u/[ √ 3 ln(L/ε)]. Plate surfaces with isotropic self-similar roughness have skin-friction coefficient: Consideration of roughness Reynolds number Re ε = 1 leads to a formula for turbulent average skinfriction coefficient from a smooth plate: and Walker [20] and Spalding and Chi [21] compiled by Churchill [11] match f σ with 0.75% RMSRE.
The discrete Fourier transform of elevations from a square portion of a rough surface can be used to determine the isotropic spatial period L P and openness Ω of that roughness. The transition from laminar or smooth turbulent flow to rough turbulent flow along a plate with isotropic, periodic roughness occurs when ε exceeds the larger of the smooth turbulent or laminar momentum thicknesses of a smooth plate at x = L P . Measurements from the present apparatus support Re λ = [0.664/ε] 2 L P L as the rough turbulent lower-bound within a factor of 2. Flow along plateau roughness (Ω < 1/2) can transition from rough to smooth turbulent flow with increasing Re x . Both plateau islands and wells were successfully modeled and tested. Flow will be rough turbulent along the entire surface when Ω > 1/2 and Re > max(Re σ , Re λ ).
Converting between local and average skin-friction coefficients differs for flows having continuous versus disrupted boundary-layers. Transforms in both directions were presented and used to compare prior work with present work. Rough turbulent forced convection is: Nu ρ = Re P r 1/3 /[6 ln 2 (L/ε)]. Bi-level plate experiments by the author match this formula with 2.2% RMSRE over the range 5000 < Re < 50000. The similarity of forced convection to natural convection from a vertical plate leads to a formula for smooth turbulent average forced convection. This formula matches the (averaged) turbulent formula from Lienhard [24] within ±6.6% over 10 5 < Re < 4.3 × 10 6 with 4.0 ≤ P r ≤ 257.
Smooth turbulent convection measurements conducted for this investigation have 1.3% RMS error relative to the present theory at 10 4 < Re < 10 5 . Over the range Re < Re λ , an isotropic, periodic plate can be treated as smooth if L/ √ ε L P > Re σ . Otherwise, laminar convection from a rough plate is Nu λ (Re), Nu σ (Re) √ 2 over the range Re < Re λ . Measurements of local skin-friction coefficient versus x/ε and Re x from the Pimenta et al sphereroughened plate match predictions of the present theory with 4.5% RMSRE. Experiments with the isotropic, periodic roughness of bi-level, sphere-roughened, mesh-covered, and perforated plates find that convection and skin-friction coefficients are close to the present theory formulas with Re as small as 5000 and as large as 5 × 10 6 .
Skin-friction and convection from such dissimilar rough surfaces hewing closely to the present theory is evidence that these formulas reflect physical laws of turbulent flow along isotropic surface roughness.
No self-similar roughness was tested in the present or prior works. The plateau roughness measurements cited or performed by this investigation were from plates with bi-level roughness.
The Colburn [28] analogies relate friction factors to forced convective heat transfer. For laminar flow: Combining equation (88) with rough f ρ equation (28) results in Nu ρ = Re P r 1/3 /[6 ln 2 (L/ε)], which is identical to rough turbulent convection equation (61). Thus, the P r dependence of laminar and rough turbulent convection are both P r 1/3 , while smooth turbulent convection's dependence is complicated in formula (69).

Conclusions
The turbulence induced in a steady flow of strength Re along a flat, smooth plate has average (not local) skin-friction coefficient f σ , average forced convection Nu σ , and local momentum thickness δ 2 : where W 0 is the principal branch of the Lambert W function: W 0 (ϕ exp ϕ) = ϕ when ϕ ≥ 0. Section 14 derived local skin-friction coefficients f σ (Re x ) and f ρ (x/ε). Periodicity combined with isotropy enables modeling of flows along many plate roughness geometries using few parameters: characteristic-length L, RMS height-of-roughness ε, isotropic spatial period L P , openness Ω, and, for plateau roughness, convex region L * . Algorithms presented find ε, L P , and Ω from an elevation grid of a square portion of the rough surface. A plate surface is isotropic, periodic roughness when its L/L P ≫ 1.
The following conclusions are for plate surfaces with L/L P ≫ 1. The average skin-friction coefficient f ρ and average forced convection Nu ρ for rough turbulence are: Fluid flow will be rough turbulent in the leading band of roughness when Re > max(Re σ , Re λ ): Fluid flow will be rough turbulent along the entire surface when Ω > 1/2 and Re > max(Re σ , Re λ ).
The following conclusions are for plate surfaces with Re σ < Re λ and L/L P ≫ 1. When Re < Re λ , a roughness with L/ √ ε L P > Re σ behaves as a smooth surface. When Re < Re λ , a roughness with L/ √ ε L P < Re σ has average forced convection: A plateau islands roughness with Ω < 1/2 and Re > Re λ has average forced convection: This Nu formula assumes ε ≪ L; a width correction for very rough plates is given in Section 23.

Nomenclature
A = rough plate area (m 2 ) c f , c f = local, average skin-friction coefficient Prandtl-Schlichting [3] C f /2, C f /2 = local, average skin-friction coefficient Mills-Hang [8] and Pimenta et al [7] C m /2 = average skin-friction coefficient Churchill [11] f c , f c = local, average skin-friction coefficient present work G(t, w) = Gray-code self-similar ramp-permutation j P = period index, the index of largest |X j | or |X j,k | k S = sand-roughness (m) L = plate characteristic-length (m) L P = roughness spatial period (m) L * = ratio of plateau convex region area to its perimeter (m) n = branching factor of profile roughness function Nu, Nu = local, average Nusselt number (convection) P r = Prandtl number of fluid q = log 2 w = positive integer Re = Reynolds number of flow parallel to the plate Re I , Re W = Re x rough-to-smooth turbulence threshold Re λ , Re σ = laminar, smooth turbulent Re upper-bound Re ε = u τ ε/ν, Re k = u τ k S /ν = roughness, sand-roughness Reynolds number Re x = x Re/L = local Reynolds number Re 0 = Re x integration lower bound S j,k = matrix of elevations t = integer u, u τ = bulk fluid, friction velocity (m/s) w = 2 q = integer power of two W (t, w) = wiggliest integer self-similar ramp-permutation W 0 = principal branch of the Lambert W function x = distance from leading edge of plate (m) X j , X j,k = discrete Fourier transform coefficient Y (t, w) = integer self-similar ramp-permutation z = roughness function (m) z = mean elevation of roughness function (m) Z = roughness random variable (m) Greek Symbols δ 2 = momentum thickness of boundary-layer flow (m) δ 2λ , δ 2σ = laminar, smooth turbulence momentum thickness (m) ǫ, ε = profile, surface RMS height-of-roughness (m) ν = fluid kinematic viscosity (m 2 /s) Ω = ratio of non-plateau area to cell area (m 2 /m 2 ) ρ = fluid density (kg/m 3 ) τ, τ 2 = fluid shearing stress (N/m 2 ) ς = peak elevation of roughness (m) ϕ, ϑ = mathematical scalar variables The goal was to measure forced convection from a rough plate over the widest practical span of Re. Figure 32 shows the rough surface of the test plate; it was milled from a slab of MIC-6 aluminum to have (676 of) square 8.33 mm × 8.33 mm × 6 mm posts spaced on 11.7 mm centers over the 30.5 cm × 30.5 cm plate. The area of the top of each post was 0.694 cm 2 , which was 50.4% of its 1.38 cm 2 cell. The RMS height-of-roughness ε = 3.00 mm. Openness Ω ≈ 49.6%. Embedded in the plate are 9 electronic resistors as heating elements and an LM35 temperature sensor. 2.54 cm of thermal insulating foam separates the back of the plate from a 0.32 mm thick sheet of aluminum with an LM35 temperature sensor at its center. Figure 33 is a cross-section illustration of the plate assembly.  Figure 35 wind-tunnel boundary-layers In order to guarantee isobaric flow, the wind-tunnel must be large enough that the boundary-layers of the test chamber and plate assembly do not interact.
The wind-tunnel test chamber in Figure 34 has a 61 cm × 35.6 cm cross-section and a 61 cm depth. This allows the plate assembly to be centered in the wind-tunnel with 15 cm of space on all sides. The fan pulling air through the test chamber produces a maximum airspeed of 4.65 m/s (Re ≈ 9.2 × 10 4 along the 30.5 cm square plate). Its minimum nonzero airspeed is 0.12 m/s (Re ≈ 2300).
The plate assembly is suspended from six lengths of 0.38 mm-diameter steel piano wire terminated at twelve zither tuning pins in wooden blocks fastened to the exterior of the test chamber. The plate is suspended face down to minimize the natural convection from the test surface. With the plate assembly in the test chamber, the airspeed increases in proportion to the reduction of test chamber aperture A e by the plate's cross-sectional area A × : Data capture and control of convection experiments are performed by an "STM32F3 Discovery 32-Bit ARM M4 72MHz" development board. The program written for the STM32F3 captures readings and writes them to the microprocessor's non-volatile RAM, controls the plate heating, servos the fan speed, and (later) uploads the captured data to a laptop computer over a USB cable. Once per second during an experiment, the program calibrates and reads each on-chip 12 bit analog-to-digital converter 16 times, summing the sixteen 12 bit readings to create a 16 bit reading per converter.
Rotations of the fan impeller are sensed by a fan blade interrupting an infrared beam. The microprocessor controls a solid-state relay (supplying power to the fan) to maintain a fan rotation rate, ω, which is dialed into switches. At ω ≤ 500 r/min, the microprocessor pulses power to the fan to phase-lock the beam interruption signal to an internal clock. At ω > 500 r/min, the microprocessor servos the duty cycle of a 10 Hz square-wave which gates power to the fan. This system operates at 32 r/min < ω < 1500 r/min.
The correspondence between fan rotation rate ω and empty test chamber airspeed u was determined using an "ABM-200 Airflow & Environmental Meter"; it specifies an accuracy of ±0.5% of reading between 2.2 m/s and 62.5 m/s. An "Ambient Weather WM-2" anemometer measured u with the rebuilt cowling.
Faster fan rotation yields diminishing increases of airspeed, suggesting formula (91), where u u is the limiting flow rate for arbitrarily fast rotation. With the parameter values it lists, Figure 36 shows the measurements, fit, and RMSRE at 800 r/min ≤ ω ≤ 1550 r/min for the original and rebuilt cowling.  Figures 37 and 38 show the fan speed variability for each downward-facing experiment; these are inputs to the computation of expected measurement uncertainty. The difference between the two figures reflects refinements of the fan-control firmware. Figure 39 shows the ambient sensor board which is at the lower edge of the test chamber in Figure 34. It measures the pressure, humidity, and temperature of the air at the wind-tunnel intake. The LM35 temperature sensor projects into the tunnel; it is wrapped in aluminum tape to minimize radiative heat transfer. To minimize self-heating, it is necessary to power the LM35 only while it is being sampled. The measurement methodology employed is unusual. Instead of waiting until the plate reaches thermal equilibrium, the plate is heated to 15 K above ambient, heating stops, the fan is started, and convection cools the plate. All of the sensor readings are captured every second during the 102 minute process, The static physical parameters from measurements and material specifications are: The effective ǫ wt may differ from the medium-density-fiberboard emissivity given by Rice [29] because the temperatures of the test chamber surfaces may not be uniform. Through the open intake, the plate also exchanges thermal radiation with objects in the room having different temperatures. Figure 34 shows duck tape applied to the lower 54% of the plate side, which corresponds to 50% coverage of the XPS foam wedge. For this partial tape coverage, ǫ W formula (92) is the area proportional mean of the duck tape emissivity and XPS emissivity. Barreira, Almeida, and Simões [30] measured emissivities of 0.86 and 0.89 from two brands of "duct tape"; the larger value is used for the aged tape on the plate sides. As of this writing, published emissivity measurements of XPS foam have not been located.
Relative to the theory in Jaffer [25], natural convection measurements (u = 0) from the plate assembly over the span of inclinations has 2.5% RMSRE when calculated with ǫ W = 0.68; the RMSRE increases to either side of 0.68. Solving ǫ W = 0.68 in equation (92) for the XPS emissivity yields ǫ XPS = 0.47, which is consistent with natural convection measurements of the plate assembly without tape.
The dynamic physical quantities which are measured every second are: u r/min fan rotation rate T F K ambient air temperature T P K plate temperature T B K back surface temperature P Pa atmospheric pressure Φ Pa/Pa air relative humidity Table 11 dynamic quantities  Table 12 lists computed quantities. U S (u) and ǫ Al ǫ wt h R A are discounted from the combined heat flow. The mean of h(u, t) over the time interval in which ∆T drops by half is the result from that experiment.
h R W/(m 2 K) radiative surface conductance U S (u) W/K side radiative and convective conductance h(u, t) W/(m 2 K) convective surface conductance Table 12 computed quantities The four sides are not isothermal; each has an exposed 3.5 mm metal strip (see cross-section Figure 40) running the length of the side, and a D W -tall wedge of extruded polystyrene foam (XPS) insulation filling the 27 mm (= √ 2 D Al ) 45 • chamfer. The local surface conductance h W (z) at elevation z (from the wedge point) is found by integrating the reciprocal distance to slab metal with respect to angle θ: Forced air flows parallel to the long dimension on two sides, but flows into the windward side and away from the leeward side. Air heated by the windward side reduces heat transfer from the test surface; air heated by the test surface suppresses heat transfer from the leeward side. Hence, the model excludes windward and leeward forced convection. The forced convective conductance of the flow-parallel foam wedges is calculated by integrating h W (z) in series (reciprocal of the sum of reciprocals, the ℓ −1 -norm) with the local surface conductance k Nu(Re x )/L, where Nu(Re x ) is the rough laminar convection calculated by applying Nu σ → Nu σ transform (71) to the rough laminar convection formula (74): The natural convection flow from the vertical faces is upward, perpendicular to the horizontal forced flow; hence, the forced convection U W and the natural convective conductance k L L ′ Nu ′ /L ′ = k L Nu ′ combine as the ℓ 2 -norm. The resulting mixed convective conductance is in mild competition with the side radiative conductance, U R = ǫ W ǫ wt h R L D W ; they combine as the ℓ Each of the four side's natural convective conductance is from the vertical plate formula (Nu ′ ) in Jaffer [25], with characteristic-length L ′ = D Al + √ 2 ε, where the metal slab thickness D Al ≈ 19.4 mm.
Collecting into U T (u) formula (96) those terms which have a factor of temperature difference T P − T F , formula (97) is the heat balance equation of the plate during convective cooling: The plate and ambient temperatures are functions of time t. Determined experimentally, the temperature group-delay through the 2.54 cm block of insulation between the slab and back sheet is 110 s: To compute Nusselt number Nu = h L/k, equation (98) is solved for the {h(u, t) A} term from equation (96).
where t ′ is the previous value of t. In equations (99, 100), T P (t), T F (t), and T B (t) are the 15-element cosine averages of plate and fluid temperatures (centered at time t).
Following Abernethy, Benedict, and Dowdell [31], the final steps in processing an experiment's data are: 1) Using equation (100), calculate the sensitivities of convected power h A ∆T per each parameter's average over the measurement time-interval; 2) multiply the absolute value of each sensitivity by its estimated parameter bias to yield component uncertainties; 3) calculate combined bias uncertainty as the root-sum-squared (RSS) of the component uncertainties; 4) calculate the RSS combined measurement uncertainty as the RSS of the combined bias uncertainty and twice the product of the rotation rate sensitivity and variability.  Tables 13 and 14 list the sensitivity, bias, and uncertainty for each component contributing more than 0.20% uncertainty for the 3 mm and 1 mm roughness plates, respectively. Figures 41 and 42 show the measurements relative to the present theory for rough turbulence and smooth turbulence, respectively.  Documentation, photographs, electrical schematics, and firmware source-code for the apparatus, as well as measurement data are available from: http://people.csail.mit.edu/jaffer/convect