Analytical Estimates of Critical Taylor Number for Motion between Rotating Coaxial Cylinders Based on Theory of Stochastic Equations and Equivalence of Measures

Abstract

The purpose of this article was to present the solution for the critical Taylor number in the case of the motion between rotating coaxial cylinders based on the theory of stochastic equations of continuum laws and the equivalence of measures between random and deterministic motions. Analytical solutions are currently of special value, as the solutions obtained by modern numerical methods require verification. At present, in the scientific literature, there are no mathematical relationships connecting the critical Taylor number with the parameters of the initial disturbances in the flow. The result of the solution shows a satisfactory correspondence of the obtained analytical dependence for the critical Taylor number to the experimental data.

1. Introduction

Analytical solutions are currently of special value, as the solutions obtained by modern numerical methods require verification. As it is known, an advantage of analytical formulas is the visualization of physical relationships between quantities. Therefore, the development of physical and mathematical theories for complex physical nonlinear processes, which are described by inhomogeneous high-order partial differential equations, is especially significant.

Moreover, analytical dependences including theoretical estimates are extremely important in the analysis of experimental data, when it is necessary to take into account the effect of substantial quantities, which are random in time and space, instead of only to give average statistical estimates.

Different ideas of the theory of turbulence are presented in [1,2,3,4,5,6,7,8,9,10]. Mathematical methods for obtaining solutions of the Navier–Stokes equation, the theory of solitons, and the theory of strange attractors are presented in [11,12,13,14,15,16,17,18,19,20,21,22,23,24].

Numerical DNS methods, and stochastic and statistical equations for investigating turbulent motion are given in [25,26,27,28,29,30,31,32,33,34]. Special attention was given to the theoretical solutions for the critical Reynolds number. It should be noted that the most well-known ratio based on the theory of dimension was determined using experimental data [35,36,37,38,39,40]. Therefore, based on these experimental formulas, it was impossible to obtain a new theory for determining analytical dependences for the critical Reynolds number of turbulence in different flows.

The theory of turbulence based on stochastic equations and the theory of equivalent measures makes it possible to derive analytical dependences for the first and second critical Reynolds numbers in the cases of isothermal and nonisothermal flows on a smooth flat plate and in a round tube [41,42,43,44]. The progress of this theory gives a new method for determining analytical dependences for profiles of averaged velocity and temperature fields [45,46,47], the friction and heat-transfer coefficients [48,49,50], second-order correlations [51,52], the correlation dimension of an attractor in the boundary layer [53,54,55,56], and the formulas for the Reynolds analogy [56,57,58] in theoretical solutions for spectral functions of the turbulent medium [59,60,61,62,63].

As a result, it was determined that the spectrum E(k)_j depends on wave number k for the interval of the generation of turbulence in the form of E(k)_j~kⁿ, n = –1.2/–1.5, –1.66 < n < –1. This formula was named as the ratio of uncertainty in turbulence generation [62,63].

Other types of fluid motions, which are important for both theory and practice, were also investigated. For example, the determination of the critical Reynolds number in a jet and for the flow near a rotating disk is presented in [64,65].

Thus, we presented here the analytical solution for the critical Taylor number in the case of the motion between rotating coaxial cylinders.

2. Equations of Conservation for the Stochastic Process

The equations were derived in [39,40,41] and take the following form:

The equation of mass (continuity)

$$\frac{d{(\rho )}_{col st}}{d\tau }=-\frac{{(\rho )}_{st}}{{\tau }_{cor}}-\frac{d{(\rho )}_{st}}{d\tau },$$

(1)

the momentum equation

$$\frac{d{\left(\rho \overrightarrow{U}\right)}_{colst}}{d\tau }={div{({\tau }_{i,j})}_{col}}_{st}+div{({\tau }_{i,j})}_{s{t}_{}}-\frac{{(\rho \overrightarrow{U})}_{st}}{{\tau }_{cor}}-\frac{d{(\rho \overrightarrow{U})}_{st}}{d\tau }+F_{col}{}_{s{t}_{}}+F_{s{t}_{}}$$

(2)

and the energy equation

$$\frac{{d{E}_{col}}_{st}}{d\tau }=div{{(\lambda \frac{\partial T}{\partial x_j}+u_i{\tau }_{i,j})}_{col}}_{st}+div{(\lambda \frac{\partial T}{\partial x_j}+u_i{\tau }_{i,j})}_{s{t}_{}}-\left(\frac{E_{st}}{{\tau }_{cor}}\right)-\left(\frac{d{E}_{st}}{d\tau }\right)+{{(u_{i}F)}_{col}}_{st}+{(u_{i}F)}_{s{t}_{}}$$

(3)

Here, $E,\rho ,\overrightarrow{U},u_i,u_j,u_l,\mu ,\tau ,{\tau }_{i,j}$ are the energy; the density; the velocity vector; the velocity components in the directions x_i, x_j, and x_l (i, j, l = 1, 2, 3), respectively; the dynamic viscosity; the time; and the stress tensor τ_i,j = P + σ_i,j, where ${\sigma }_{i,j}=\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)-{\delta }_{ij}\left(\xi -\frac{2}{3}\mu \right)\frac{\partial u_l}{\partial x_l}$, i and j are the tensorial notations, δ_ij = 1 if I = j, δ_ij = 0 for i$\ne$j. j is the pressure of a liquid or gas; $\lambda$ is the thermal conductivity; $c_p$ and $c_v$ are the specific heat at constant pressure and volume, respectively; F is the external force. Further, L = L_U,P = L_U is the scale of turbulence. The subscripts (_U,P) and (_U) refer to the velocity field and the subscript (_T) refers to the temperature field. The turbulence scale L is taken along the radius r. The subscript “col st” refers to the components that are deterministic. The subscript “st” refers to the components that are stochastic. Then, for the nonisothermal motion of the medium, using the definition of equivalence of measures between deterministic and random processes at the critical point, the sets of stochastic equations of energy, momentum, and mass are defined for the next space–time areas: (1) the onset of generation (subscript 1,0 or 1); (2) the generation (subscript 1,1); (3) the diffusion (1,1,1); and (4) the dissipation of the turbulent fields.

3. Stochastic Equations for Critical Taylor Number

For the critical numbers, sets (1)–(3) of the equations of mass, momentum, and energy for the area (1) referring to the pair (N, M) = (1, 0) is:

$${\left(\frac{d{(\rho )}_{col,st}}{d\tau }\right)}_{1,0}=-\frac{{\rho }_{st}}{{\tau }_{cor}},$$

$$\{\begin{array}{l}{\left(\frac{d{(\rho \overrightarrow{U})}_{col,st}}{d\tau }\right)}_{1,0}=-\left(\frac{{(\rho \overrightarrow{U})}_{st}}{{\tau }_{cor}}\right);\\ div{({\tau }_{i,j})}_{col,st1}=\frac{{(\rho \overrightarrow{U})}_{st}}{{\tau }_{cor}},\end{array}$$

(4)

$$\{\begin{array}{l}{\left(\frac{d{(E)}_{col,st}}{d\tau }\right)}_{1,0}=-{\left(\frac{{(E)}_{st}}{{\tau }_{cor}}\right)}_{1,0};\\ div{(\lambda \frac{\partial T}{\partial x_j}+u_i{\tau }_{i,j})}_{col,st1}={\left(\frac{{(E)}_{st}}{{\tau }_{cor}}\right)}_{1,0}\end{array}.$$

The motion between rotating coaxial cylinders is described in detail in [34,35,36,37,38]. In particular, for the laminar motion during the rotation of both cylinders, the velocity profile is determined for laminar motion by the dependence [31]

$$u_r=\left[\frac{1}{r_2^2-r_1^2}\right]\left[r\left({\omega }_2{r}^2{}_2-{\omega }_1{r}^2{}_1\right)-\frac{r^2{}_2{r}^2{}_1}{r}({\omega }_2-{\omega }_1)\right].$$

(5)

In the case when the inner cylinder rotates and the outer cylinder is at rest, it is also characterized by the bifurcation in laminar motion with the formation of Taylor vortices instead of only by the transition of deterministic motion to the random motion. Then, the velocity profile for the laminar motion without the formation of Taylor vortices [31] is

$$u_r=\left[\frac{1}{d\left(r_2^{}+r_1^{}\right)}\right]\left[\frac{r^2{}_2{r}^2{}_1}{r}({\omega }_1)-r\left({\omega }_1{r}^2{}_1\right)\right].$$

(6)

In Formulas (5) and (6), r₁, r₂, ϖ₁, ϖ₂, and d are the radius and speed of rotation of the first and second cylinders, and the distance between the cylinders, respectively.

Taking into account the set of stochastic equations of the presented theory of equivalent measures for a continuous isothermal medium, we write for the space–time region of the beginning of the resonance–correlation “determinism–randomness” according to the equivalence of measures (4) of deterministic and random motion, and we define the derivative

$$\frac{\partial u_r}{\partial r}=\frac{\partial }{\partial r}\left[\frac{{\omega }_1\left(r^2{}_1\right)}{d\left(r_2^{}+r_1^{}\right)}\right]\left[\frac{r^2{}_2}{r}-r\right]=\left[\frac{{\omega }_1\left(r^2{}_1\right)}{d\left(r_2^{}+r_1^{}\right)}\right]\left[-\frac{r^2{}_2}{r^2}-1\right]$$

$$\frac{\partial u_r}{\partial r}=\frac{\partial }{\partial r}\left[\frac{{\omega }_1\left(r^2{}_1\right)}{d\left(r_2^{}+r_1^{}\right)}\right]\left[\frac{r^2{}_2}{r}-r\right]=\left[\frac{{\omega }_1\left(r^2{}_1\right)}{d\left(r_2^{}+r_1^{}\right)}\right]\left[-\frac{r^2{}_2}{r^2}-1\right]$$

$$\mu {\left(\frac{\partial u_r}{\partial r}\right)}^2=\rho \cdot \nu \cdot {\left[\frac{{\omega }_1\left(r^2{}_1\right)}{d\left(r_2^{}+r_1^{}\right)}\right]}^2{\left[\frac{r^2{}_2}{r^2}+1\right]}^2=\rho \cdot \nu \cdot {\left[\frac{{\omega }_1}{d}\right]}^2{\left[\frac{\left(r^2{}_1\right)}{\left(r_2^{}+r_1^{}\right)}\right]}^2{\left[\frac{r^2{}_2+r^2}{\left(r_{}^2\right)}\right]}^2$$

(7)

Transforming Equation (7), we obtain

$$\mu {\left(\frac{\partial u_r}{\partial r}\right)}^2=\rho \cdot \nu \cdot {\omega }_1{}^2\left[\frac{\left(r^4{}_1\right)}{d^2}\right]{\left[\frac{1}{r_1\left(2+d/r_1^{}\right)}\right]}^2{\left[\frac{r^2{}_2+r^2}{\left(r_{}^2\right)}\right]}^2\approx \rho \cdot \nu \cdot {\omega }_1{}^2\left[\frac{\left(r^2{}_1\right)}{d^2}\right].$$

(8)

then

$$\mu {\left(\frac{\partial u_r}{\partial r}\right)}^2\approx \rho \cdot \nu \cdot {\omega }_1{}^2\left[\frac{\left(r^2{}_1\right)}{d^2}\right]={\left|\frac{E_{st}}{{\tau }_{cor}^0}\right|}_{1,0}$$

(9)

or we have $\left(\frac{\nu }{d^2}{\tau }_{cor}^0\right)\left(\frac{r^2{}_1{\omega }_1{}^2}{E_{st}/\rho }\right)=1$.

For the case of the time correlation ${\left({\tau }_{cor}^0\right)}_{1,0}=\frac{L}{\sqrt{E_{st}/\rho }}$, we can write

$$\left(\frac{\nu }{d^2}\frac{L}{\sqrt{E_{st}/\rho }}\right)\left(\frac{r^2{}_1{\omega }_1{}^2}{E_{st}/\rho }\right)=1$$

or

$$\left(\left(\frac{\nu }{r^{}{}_1{\omega }_1{}^{}{d}^{}}\right)\frac{L}{d}\right){\left(\frac{r^{}{}_1{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^3=1$$

(10)

Thus, we write that

$$\left(\frac{r^{}{}_1{\omega }_1{}^{}{d}^{}}{\nu }\right)=\left(\frac{r_1}{d}\frac{L}{r_1}\right){\left(\frac{r^{}{}_1{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^3$$

(11)

and obtain the dependence

$$Ta=\left(\frac{r^{}{}_1{\omega }_1{}^{}{d}^{}}{\nu }\right)\sqrt{\frac{d}{r_1{}^{}}}=\left(\sqrt{\frac{r_1}{d}}\left(\frac{L}{r_1}\right)\right){\left(\frac{r_1{}^{}{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^3$$

(12)

and, finally, the formula for the critical Taylor’s number is

$$Ta=\left(\frac{r^{}{}_1{\omega }_1{}^{}{d}^{}}{\nu }\right)\sqrt{\frac{d}{r_1{}^{}}}=\left(\sqrt{\frac{r_1}{d}}\left(\frac{L}{d}\right)\left(\frac{d}{r_1}\right)\right){\left(\frac{r_1{}^{}{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^3=\left(\sqrt{\left(\frac{d}{r_1}\right)}\right){\left(\frac{r_1{}^{}{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^3\left(\frac{L}{d}\right)$$

(13)

4. Critical Point in the Case of Motion between Rotating Coaxial Cylinders

We determine the position of the critical point. The critical point definition is found from the equation

$${\displaystyle \underset{-\Delta V|2}{\overset{+\Delta V|2}{\int }}}d{\left(E_{co{l}_{st}}\right)}_{1;0}={\displaystyle \underset{X}{\int }d{E}_{st}}$$

(14)

$E_{st}$ is the random energy component in the space $X$ with the measure $m\left(E_{st}\right)<\infty$

$$E_{st}=E_{st}(\overrightarrow{x_i},{\tau }_i,m_i)<\infty$$

(15)

In accordance with ergodic theory,

$${\displaystyle \underset{X}{\int }d{E}_{st}}=\frac{1}{\Delta V}{\displaystyle \underset{V}{\int }{E}_{st}\delta \left({(\Delta V)}_{critic}-\Delta V\right)}dV=\frac{1}{{\tau }_{cor}^0}{\displaystyle \underset{\tau }{\int }{E}_{st}\delta ({\tau }_{cor}^0-\tau )}d\tau ={(E_{st})}_{critic}$$

(16)

${(E_{st})}_{critic}$ is the energy of the stochastic field at the critical point, or

$${\displaystyle \underset{X}{\int }d{E}_{st}}=\frac{1}{L}{\displaystyle \underset{L}{\int }{E}_{st}\delta \left({(x_i)}_{critic}-x_i\right)}dL=\frac{1}{{\tau }_{cor}^0}{\displaystyle \underset{\tau }{\int }{E}_{st}\delta ({\tau }_{cor}^0-\tau )}d\tau ={\left(E_{st}\right)}_{critic}$$

(17)

L is the scale of disturbance. Then, we can write

$$\begin{gathered} {\displaystyle \underset{-V|2}{\overset{+V|2}{\int }}}d{\left(E_{co{l}_{st}}\right)}_{1;0}\cong {\left(0,5\rho u^2{}_r\right)}_{-L/2}^{+L/2}=0,5\rho {\omega }_1{}^2{{\left(\left[\frac{\left(r^2{}_1\right)}{d\left(r_2^{}+r_1^{}\right)}\right]\left[\frac{r^2{}_2}{r}-r\right]\right)}^{2 }}_{-L/2}^{+L/2}= \\ \begin{array}{l}=0.5\rho {\omega }^2{\left[\frac{r^2{}_1}{d\left(r_2^{}+r_1^{}\right)}\right]}^2\left\{{\left[\frac{r^2{}_2}{(r+L/2)}-(r+L/2)\right]}^2-{\left[\frac{r^2{}_2}{(r-L/2)}-(r-L/2)\right]}^2\right\}\approx 0.5\rho {\omega }^2{\left[\frac{r^2{}_1}{d\left(r_2^{}+r_1^{}\right)}\right]}^2\frac{3.75d^2{r}^2{}_{2}2rL}{{(r+L/2)}^2{(r-L/2)}^2}\\ \end{array} \end{gathered}$$

(18)

An estimate of Equation (18), when $r\to r_1$ and $r_2\ge r_1>>d$, may be written as

$$0.5\rho {\omega }_1{}^2{\left[\frac{r^2{}_1}{d\left(r_2^{}+r_1^{}\right)}\right]}^2\frac{3.75d^2{r}^2{}_{2}2rL}{{(r+L/2)}^2{(r-L/2)}^2}=0.5\rho {\omega }_1{}^2{\left[\frac{r^2{}_1}{2d{r}_2}\right]}^2\frac{3.75d^2{r}^2{}_{2}2rL}{{(r+L/2)}^2{(r-L/2)}^2}\approx K\cdot \rho \left({\omega }_1{}^2{r}_1^2\right)\frac{L}{r}$$

(19)

Here, K = 0.94/0.93, and finally,

$${\displaystyle \underset{-V|2}{\overset{+V|2}{\int }}}d{\left(E_{co{l}_{st}}\right)}_{1;0}\approx K\rho {\omega }_1{}^2{r}^2{}_1\left(\frac{L}{d}\right)\left(\frac{d}{r_1}\right)\frac{r_1}{r_{}}$$

(20)

Therefore, we obtain

$$K\rho {\omega }_1{}^2{r}^2{}_1\left(\frac{L}{d}\right)\left(\frac{d}{r_1}\right)\frac{r_1}{r_{}}\approx E_{st}^{}$$

(21)

Then, we finally have the formula for the critical point in the case of the motion between rotating coaxial cylinders

$$r_{cr}\approx K\left(\frac{d}{r_1}\right){\left(\frac{r_1{}^{}{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^2\left(\frac{L}{d}\right)r_1$$

(22)

5. Results of Estimates of the Critical Taylor Number

We present the results of calculations for the conditions of the experiment of Taylor [31], when (d/r₁) = 0.028, the pulsation intensity $\left(\frac{\sqrt{E_{st}/\rho }}{r_1{}^{}{\omega }_1{}^{}}\right)$ is 1–2%, the relative magnitude of the turbulence scale (L/d) is 0.01–0.02, and the experimental value of the critical Taylor number is (Ta)cr~400 [31,66].

We emphasize that it is of interest to make a similar estimate, when the laminar-motion velocity profile is determined by the motion of Taylor vortices, but this version of the definition is not used here.

Thus, the critical Taylor number and critical point using Formulas (13) and (22), and depending on the values of the pulsation intensity and the scale determined from experiments, may be calculated as

$$Ta=\left(\sqrt{\left(\frac{d}{r_1}\right)}\right){\left(\frac{r_1{}^{}{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^3\left(\frac{L}{d}\right)=\left(\sqrt{\left(0.028\right)}\right){\left(62.\right)}^3\left(0.01\right)=398$$

(23)

$$r_{cr}\approx 0.93\left(\frac{d}{r_1}\right){\left(\frac{r_1{}^{}{\omega }_1{}^{}}{\sqrt{E_{st}/\rho }}\right)}^2\left(\frac{L}{d}\right)r_1=0.93{\left(62.\right)}^2\left(0.01\right)\left(0.028\right)r_1=1.0009r_1$$

(24)

The accuracy and the physical validity of the expression for the critical point can be verified indirectly. Thus, having experimental relationships for quantities $\left(\frac{d}{r_1}\right),\left(\frac{L}{d}\right)$, it is possible to obtain an indirect empirical value for the scale of turbulence. At the same time, calculating the distance from the surface of the first cylinder to the critical point $\Delta r=r_{cr}-r_1$, it is necessary to take the ratio of the scale to this distance. The resulting value must be compared with the von Karman constant.

Thus, $\left(\frac{d}{r_1}\right)=0.028,\left(\frac{L}{d}\right)=0.01$; thus, $L=0.00028r_1$. The distance from the surface of the first cylinder to the critical point is $\Delta r=r_{cr}-r_1\approx 0.0009r_1$. The calculated value is k ~$L/\Delta r\approx 0.312$ and, at the same time, the empirical value of the von Karman constant is k = 0.39/0.41. Thus, the deviation in the calculated dependence from the empirical one is less than 25%, which confirms the satisfactory agreement of dependence (22) for the critical point. It should be noted that the main studies [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125] do not contain information on analytical solutions for the critical Taylor number as a function of the initial intensity and scale of perturbation.

6. Conclusions

We presented the analytical Formula (13) for the critical Taylor number for the motion between rotating coaxial cylinders based on the theory of stochastic equations of continuum laws and the equivalence of measures between random and deterministic motions. The analytical Formula (22) for the critical point in the case of the motion between rotating coaxial cylinders is also derived. The result of the solution of Equation (13) shows a satisfactory correspondence with the obtained analytical dependence for the critical Taylor number to the experimental data. For validating the dependence for the critical point, the indirect calculated dependence of the von Karman constant was compared with the empirical value of this constant. The result of the comparison shows the satisfactory agreement between the calculated and empirical values of the von Karman constant.

Thus, the obtained analytical dependences can be used for validating both experimental and numerical studies of the onset of turbulence in a fluid flow between rotating cylinders, which is important for the practice of developing various technical devices such as hydraulic cylinders, as well as oil and water-cooling systems.