# Numerical Simulations of Wave-Induced Flow Fields around Large-Diameter Surface-Piercing Vertical Circular Cylinder

## Abstract

**:**

## 1. Introduction

## 2. Wave Regimes

#### 2.1. Overview of Wave Regimes

**Figure 1.**Scheme of wave regimes in terms of KC and $D/L$: (××) waves selected for a preliminary evaluation of the quantities defined in Equations (10), (11), (14) and (15); (∆∆) parameters of the experimental waves [14] falling under the limit mirrored by Equation (5).

#### 2.2. Viscous-Fluid Wave Framework

**Figure 2.**Scheme of viscous-fluid oscillatory-flow regimes in terms of KC and $\mathrm{Re}$: (××) waves selected for a preliminary evaluation of the quantities defined in Equations (10), (11), (14) and (15); (∆∆) parameters of the experimental waves [14].

## 3. Numerical Techniques

#### 3.1. Velocity-Potential Solution

D/L | KC | $H/{r}_{0}$ | d/L | kd | $k{r}_{0}$ |
---|---|---|---|---|---|

0.2–0.4 | 0.10–0.76 | 0.480–0.064 | 0.40–0.80 | 2.51–5.03 | 0.63–1.26 |

#### 3.2. Numerical Integration of the Euler Equations

Field Parameters | ||||||

KC | D [m] | H [m] | L [m] | T [s] | ||

0.70 | 12.0 | 2.65 | 45.11 | 5.41 | ||

d [m] | ${r}_{0}$ [m] | $D/L$ | $H/{r}_{0}$ | $H/L$ | ||

18.0 | 6.0 | 0.266 | 0.44 | 0.059 | ||

${\left(H/L\right)}_{Max1}$ | ${\left(H/L\right)}_{Max2}$ | $d/L$ | kd | $k{r}_{0}$ | ||

0.138 | 0.069 | 0.40 | 2.51 | 0.84 | ||

Computational Parameters | ||||||

${L}_{x}$=${L}_{y}$ [m] | ${L}_{z}$ [m] | ${N}_{x}$ = ${N}_{y}$ | ${N}_{z}$ | ${N}_{tot}$ | $\text{\Delta}x$=$\text{\Delta}y$ [m] | $\text{\Delta}z$ [m] |

90.22 | 36.0 | 450 | 360 | $72.90\times {10}^{6}$ | 0.20 | 0.10 |

#### 3.3. Numerical Integration of the Navier-Stokes Equations

#### 3.3.1. Validation of the Navier-Stokes Solver

**Figure 4.**Computational test-case. Mean streamwise-velocity profile: (—) reference solver, (●●) Flow-3D solver, (―) law of the wall ${u}^{+}={y}^{+}$, ${u}^{+}=2.5\mathrm{ln}{y}^{+}+5.5$.

Domain Dimensions ($-1<y/h<+1$, $-200<{y}^{+}<+200$) | |||||

${L}_{x}$ | ${L}_{y}$ | ${L}_{z}$ | ${L}_{x}^{+}$ | ${L}_{y}^{+}$ | ${L}_{z}^{+}$ |

$4\pi h$ | $2h$ | $2\pi h$ | 2512 | 400 | 1256 |

Computational Grid for the Reference Navier-Stokes Solver | |||||

${N}_{x}$ | ${N}_{y}$ | ${N}_{z}$ | ${N}_{tot}$ | ||

256 | 181 | 256 | $\approx 11.862\times {10}^{6}$ | ||

Multi-Bolck Computational Grid for the Flow-3D Navier-Stokes Solver | |||||

Block-subdivision along y | ${N}_{x}$ | ${N}_{y}$ | ${N}_{z}$ | ${N}_{tot}$ | |

$-200<{y}^{+}<-198.54$ | 3040 | 5 | 1520 | $\approx $ 23.1$\times {10}^{6}$ | |

$-198.54<{y}^{+}<-191.88$ | 1520 | 13 | 760 | ≈ 15.1$\times {10}^{6}$ | |

$-191.88<{y}^{+}<-167.72$ | 760 | 18 | 380 | ≈ 5.2$\times {10}^{6}$ | |

$-167.72<{y}^{+}<+167.72$ | 380 | 107 | 190 | ≈ 7.7$\times {10}^{6}$ | |

$+167.72<{y}^{+}<+191.88$ | 760 | 18 | 380 | ≈ 5.2$\times {10}^{6}$ | |

$+191.88<{y}^{+}<+198.54$ | 1520 | 13 | 760 | ≈ 15.1$\times {10}^{6}$ | |

$+198.54<{y}^{+}<+200$ | 3040 | 5 | 1520 | ≈ 23.1$\times {10}^{6}$ | |

$-200<{y}^{+}<+200$ | 11020 | 179 | 5510 | ≈ 94.4$\times {10}^{6}$ |

**Figure 5.**Computational test-case. Rms values of the velocity fluctuations; ${{u}^{\prime}}_{rms}$: (—) reference solver, (●●) Flow-3D solver; ${{v}^{\prime}}_{rms}$: (— — ) reference solver, (

**♦♦**) Flow-3D solver; ${{w}^{\prime}}_{rms}$: (– – ) reference solver, (▲▲) Flow-3D solver.

**Figure 6.**Computational test-case. Reynolds shear stress; $-\overline{{u}^{\prime}{v}^{\prime}}$: (—) reference solver, (●●) Flow-3D solver.

**Figure 7.**Computational test-case. Skewness factors of the velocity fluctuations; ${S}_{{u}^{\prime}}$: (—) reference solver, (●●) Flow-3D solver; ${S}_{{v}^{\prime}}$: (— — ) reference solver, (

**♦♦**) Flow-3D solver; ${S}_{{w}^{\prime}}$: (– – ) reference solver, (▲▲) Flow-3D solver.

**Figure 8.**Computational test-case. Flatness factors of the velocity fluctuations; ${F}_{{u}^{\prime}}$: (—) reference solver, (●●) Flow-3D solver; ${F}_{{v}^{\prime}}$: (— —) reference solver, (

**♦♦**) Flow-3D solver; ${F}_{{w}^{\prime}}$: (– –) reference solver, (▲▲) Flow-3D solver.

#### 3.3.2. Accuracy of Calculations and Computing Procedures

Field Parameters ($KC=0.578$, $\mathrm{Re}=1862$, $\beta =\mathrm{Re}/KC=3221$) | |||||

D | L | H | a | d | ${r}_{0}$ |

[D units] | [D units] | [D units] | [D units] | [D units] | [D units] |

1 | 3.906 | 0.183 | 0.0915 | 2 | 0.50 |

$\eta $ | T | ${\tau}_{\eta}$ | ${u}_{\mathrm{max}}$ | $D/L$ | $H/{r}_{0}$ |

[D units] | [$\sqrt{D/g}$ units] | [$\sqrt{D/g}$ units] | [$\sqrt{Dg}$ units] | ||

0.0147 | 4.962 | 3.450 | 0.116 | 0.256 | 0.367 |

$H/L$ | ${\left(H/L\right)}_{Max1}$ | ${\left(H/L\right)}_{Max2}$ | $d/L$ | kd | $k{r}_{0}$ |

0.047 | 0.139 | 0.0696 | 0.512 | 3.217 | 0.809 |

Computational Parameters [D Units] | |||||

${L}_{x}\left(=2L\right)$ | ${L}_{y}\left(=2L\right)$ | ${L}_{z}\left(=3d\right)$ | ${L}_{1}={L}_{2}={L}_{4}={L}_{5}$ | ${{L}^{\prime}}_{3}$ | |

7.8 | 7.8 | 3.0 | 1.3 | 2.1 | |

block 1 | $-3.9<x<-2.6$ | $-3.9<y<3.9$ | $-2<z<1.0$ | ${N}_{block\_1}=$1,156,200 | |

block 2 | $-2.6<x<2.6$ | $-3.9<y<-2.6$ | $-2<z<1.0$ | ${N}_{block\_2}=$702,240 | |

block 3 | $-2.6<x<2.6$ | $-2.6<y<2.6$ | $-2<z<1.0$ | ${N}_{block\_3}=$14,953,080 | |

block 4 | $-2.6<x<2.6$ | $2.6<y<3.9$ | $-2<z<1.0$ | ${N}_{block\_4}=$702,240 | |

block 5 | $2.6<x<3.9$ | $-3.9<y<3.9$ | $-2<z<1.0$ | ${N}_{block\_5}=$1,156,200 | |

blocks 1-5 | $-3.9<x<3.9$ | $-3.9<y<3.9$ | $-2<z<1.0$ | ${N}_{tot}=$18,669,960 | |

Space and Time Resolutions [ $\eta $ and ${\tau}_{\eta}$ Units] | |||||

$\text{\Delta}{x}_{cylinder\_wall}=0.45\eta $ | $\text{\Delta}{x}_{end\_of\_block\_3}=2.25\eta $ | $\text{\Delta}{x}_{blocks\_1,5}=2.25\eta $ | |||

$\text{\Delta}{z}_{blocks\_1,2,3,4,5}=1.70\eta $ | $\text{\Delta}t=0.002{\tau}_{\eta}$ |

## 4. Flow-Field Analysis

#### 4.1. Overview of Swirling-Strength Criterion for Flow-Structure Extraction

#### 4.2. Overview of Karhunen-Loève Decomposition Technique

Instant Number ($t$) | Instant ($\sqrt{D/g}$ Units) | Instant ($T$ Units) |
---|---|---|

1 | 0.000 | 0.000 |

2 | 0.876 | 0.144 |

3 | 2.481 | 0.500 |

4 | 3.722 | 0.750 |

5 | 4.962 | 1.000 |

Mode Number | Energy Fraction | Energy Sum |
---|---|---|

1 | 0.87818 | 0.87818 |

2 | 0.10770 | 0.98588 |

3 | 1.2271$\times {10}^{-2}$ | 0.99815 |

4 | 5.0327$\times {10}^{-3}$ | 1.00 |

5 | 2.6852$\times {10}^{-3}$ | 1.00 |

6 | 9.2960$\times {10}^{-4}$ | 1.00 |

7 | 5.5663$\times {10}^{-6}$ | 1.00 |

8 | 1.2803$\times {10}^{-6}$ | 1.00 |

9 | 3.5057$\times {10}^{-7}$ | 1.00 |

10 | 2.1537$\times {10}^{-7}$ | 1.00 |

## 5. Results

#### 5.1. Forces and Runups

#### 5.2. Velocity Potential

**Figure 12.**Velocity-potential-derived free-surface profile. Top view of computing domain, at: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{\phi}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{\phi}$.

**Figure 13.**Velocity-potential-derived free-surface profile. Side view of computing domain, at: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{\phi}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{\phi}$.

#### 5.3. Euler Equations

**Figure 14.**Flow structures as obtained from the solution of the primitive-variable Euler equations. Top view of free-surface profile: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 15.**Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Upstream view: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 16.**Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Side view: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

**Figure 17.**Close-up at the cylinder surface of inviscid-flow structures as obtained from the solution of the Euler equations. Downstream view: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{Euler}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{Euler}$.

#### 5.4. Navier Stokes Equations

**Figure 18.**Flow structures as obtained from the solution of the primitive-variable Navier-Stokes equations. Top view of free-surface profile: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 19.**Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Upstream view: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 20.**Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Side view: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 21.**Close-up at the cylinder surface of viscous-flow structures as obtained from the solution of the Navier-Stokes equations. Downstream view: (

**a**) $t={t}_{{F}_{\mathrm{max}}}^{NS}$; (

**b**) $t={t}_{{R}_{\mathrm{max}}}^{NS}$.

**Figure 22.**Close-up views of structures as obtained from the solution of the Navier-Stokes equations. Bottom view of computing domain at $t={t}_{{F}_{\mathrm{max}}}^{NS}$. The external surfaces of structures are colored with: (

**a**) wave-field spanwise vorticity; (

**b**) wave-field pressure values.

**Figure 23.**Close-up views of structures as obtained from the solution of the Navier-Stokes equations. Bottom view of computing domain at $t={t}_{{R}_{\mathrm{max}}}^{NS}$. The external surfaces of structures are colored with: (

**a**) wave-field spanwise vorticity; (

**b**) wave-field pressure values.

**Figure 24.**Flow-field representation at $T=0.000$ (T is wave period), top view (free-surface level): (

**a**) flow-structures (colored with spanwise vorticity); (

**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 25.**Flow-field representation at $T=0.144$ (T is wave period), top view (free-surface level): (

**a**) flow-structures (colored with spanwise vorticity); (

**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 26.**Flow-field representation at $T=0.500$ (T is wave period), top view (free-surface level): (

**a**) flow-structures (colored with spanwise vorticity); (

**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 27.**Flow-field representation at $T=0.750$ (T is wave period), top view (free-surface level): (

**a**) flow-structures (colored with spanwise vorticity); (

**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

**Figure 28.**Flow-field representation at $T=1.000$ (T is wave period), top view (free-surface level): (

**a**) flow-structures (colored with spanwise vorticity); (

**b**) surfaces of constant x-velocity from reconstruction based on the three most energetic KL modes.

## 6. Concluding Remarks

## Conflicts of Interest

## Nomenclature

## Roman Symbols (Upper Case)

${A}_{ij}$ | velocity-gradient tensor |

$D$ | cylinder diameter |

Dsc | discriminant of characteristic equation |

${F}_{f}$ | portion-due-to-friction of force on cylinder |

${F}_{\mathrm{max}}$ | maximum force on cylinder |

${F}_{\mathrm{max}}^{nd}$ | maximum force on cylinder (nondimensional) |

H | wave height |

${H}_{p}^{\left(1\right)}$ | Hankel function of first kind (order p) |

${{H}^{\prime}}_{p}^{\left(1\right)}$ | first derivative of ${H}_{p}^{\left(1\right)}$ |

${J}_{p}$ | Bessel function of first kind (order p) |

${{J}^{\prime}}_{p}$ | first derivative of ${J}_{p}$ |

KC | Keulegan-Carpenter number |

$L$ | wave length |

${L}_{x},{L}_{y},{L}_{z}$ | dimensions of computing domain along (x,y,z) |

${L}_{x}^{+},{L}_{y}^{+},{L}_{z}^{+}$ | dimensions of computing domain along (x,y,z) in wall units (test case) |

${N}_{x},{N}_{y},{N}_{z}$ | number of grid points in the computing domain along (x,y,z) |

${N}_{tot}$ | total number of grid points in the computing domain |

P,Q,R | scalar invariants of velocity-gradient tensor |

PP | parameter in hyperbolic-tangent grid-stretching law |

parameter in hyperbolic-tangent grid-stretching law | |

${R}_{\mathrm{max}}$ | maximum runup at cylinder surface |

${R}_{\mathrm{max}}^{nd}$ | maximum runup at cylinder surface (nondimensional) |

$\mathrm{Re}$ | wave-field Reynolds number |

${\mathrm{Re}}_{NS}$ | Reynolds number resulting from nondimensionalization of Navier-Stokes equations |

${\mathrm{Re}}_{\tau}$ | friction-velocity Reynolds number (test case) |

$T$ | wave period |

${Y}_{p}$ | Bessel function of second kind (order p) |

${{Y}^{\prime}}_{p}$ | first derivative of ${Y}_{p}$ |

## Roman Symbols (Lower Case)

a | wave amplitude |

d | still-water level |

g | acceleration due to gravity ($g=9.807$ $m/{s}^{2}$) |

h | channel half-width (test case) |

k | wavenumber |

p | pressure (also index) |

real | real part of complex quantity |

r,$\theta $,z | cylindrical coordinate system |

${r}_{0}$ | cylinder radius |

t | time coordinate |

${t}_{{F}_{max}}^{Euler}$ | time at which the Euler-equations-derived ${F}_{\mathrm{max}}$ verifies |

${t}_{{R}_{max}}^{Euler}$ | time at which the Euler-equations-derived ${R}_{\mathrm{max}}$ verifies |

${t}_{{F}_{\mathrm{max}}}^{NS}$ | time at which the Navier-Stokes-equations-derived ${F}_{\mathrm{max}}$ verifies |

${t}_{{R}_{\mathrm{max}}}^{NS}$ | time at which the Navier-Stokes-equations-derived ${R}_{\mathrm{max}}$ verifies |

${t}_{{F}_{\mathrm{max}}}^{\phi}$ | time at which the $\phi $-derived ${F}_{\mathrm{max}}$ verifies |

${t}_{{R}_{\mathrm{max}}}^{\phi}$ | time at which the $\phi $-derived ${R}_{\mathrm{max}}$ verifies |

${u}_{i}\left(u,v,w\right)$ | velocity components along (x,y,z) |

${{u}^{\prime}}_{i}\left({u}^{\prime},{v}^{\prime},{w}^{\prime}\right)$ | fluctuating-velocity components along (x,y,z) (test case) |

${u}_{\mathrm{max}}$ | maximum value of $u$ |

${x}_{i}\left(x,y,z\right)$ | cartesian coordinates (x is wave direction, z is vertical direction) |

## Greek Symbols (Upper Case)

$\text{\Delta}t$ | time resolution of calculations |

$\text{\Delta}x,\text{\Delta}y,\text{\Delta}z$ | space resolution of calculations along (x,y,z) |

## Greek Symbols (Lower Case)

$\beta $ | $\mathrm{Re}/KC$ |

$\delta $ | phase angle |

${\delta}_{ij}$ | Kronecker delta |

$\overline{\epsilon}$ | average rate of dissipation of kinetic energy per unit mass |

${\epsilon}_{ijk}$ | alternating-unit tensor |

$\lambda $ | eigenvalue |

${\lambda}_{r}$ | real eigenvalue |

${\lambda}_{cr}$ | real part of complex eigenvalue |

${\lambda}_{ci}$ | imaginary part of complex eigenvalue pair (swirling strength) |

${\left({\lambda}_{ci}\right)}_{th}$ | threshold value of swirling strength |

$\eta $ | Kolmogorov length scale |

$\nu $ | fluid kinematic viscosity |

${\tau}_{w}$ | mean shear stress at the wall |

${\tau}_{\eta}$ | Kolmogorov time scale |

$\phi $ | velocity-potential |

$\rho $ | fluid density |

$\omega $ | wave angular frequency |

${\omega}_{i}$ | vorticity vector |

${\omega}_{y}$ | spanwise component of vorticity |

## Acronyms (Upper Case)

ADI | Alternating Direction Implicit (scheme) |

DNS | Direct Numerical Simulation (technique) |

FAVOR | Fractional Area Volume Obstacle Representation (technique) |

GMRES | Generalized Minimal Residual (method) |

KL | Karhunen-Loève (decomposition technique) |

RANS | Reynolds Averaged Navier-Stokes (equations) |

RNG | Renormalization Group (theory) |

VOF | Volume Of Fluid (method) |

## Acronyms (Lower Case)

lhs | left-hand side (of equation) |

rhs | right-hand side (of equation) |

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**MDPI and ACS Style**

Alfonsi, G.
Numerical Simulations of Wave-Induced Flow Fields around Large-Diameter Surface-Piercing Vertical Circular Cylinder. *Computation* **2015**, *3*, 386-426.
https://doi.org/10.3390/computation3030386

**AMA Style**

Alfonsi G.
Numerical Simulations of Wave-Induced Flow Fields around Large-Diameter Surface-Piercing Vertical Circular Cylinder. *Computation*. 2015; 3(3):386-426.
https://doi.org/10.3390/computation3030386

**Chicago/Turabian Style**

Alfonsi, Giancarlo.
2015. "Numerical Simulations of Wave-Induced Flow Fields around Large-Diameter Surface-Piercing Vertical Circular Cylinder" *Computation* 3, no. 3: 386-426.
https://doi.org/10.3390/computation3030386