numerical computation of a free surface flow around a

9

(Read at the Antumn Meeting of The Society of Naval Architects of Japan, Nov. 1988)

Numerical Computation of a Free Surface Flow

around a Submerged Hydrofoil

by the Euler/Navier-Stokes Equations

by Takanori Hino *, Member

Summary

The finite difference methods for the 2-dimensional Navier-Stokes and the Euler equations

with the nonlinear free surface conditions are developed to study the basic characteristics of

the numerical scheme. The numerical experiments are made for the free surface flow around

the hydrofoil of the NACA 0012 wing section. The effect of the difference of the numerical

implementation of the boundary conditions for the Euler solver on the final solutions are

discussed. The Euler solution with the proper boundary condition is compared with the

Navier-Stokes solution together with the experiment or other inviscid calculations.

1. Introduction

Various finite difference methods for the Navier-Stokes equations have been developed and have demonstrated their possibility to solve viscous flow

problems in the field of ship hydrodynamics1)-5). Because the viscous flow fields governed by the Navier-Stokes equations are nonlinear and compli-cated in general it is not easy to validate numerical solutions. The convergence of schemes can be verified by checking the effects of a grid spacing, a solution domain, a time increment and so forth. However, this cannot tell whether the converged solution of the scheme is physically reasonable or not. The detailed comparison of a solution with the exact solution is required for this purpose. Because the exact solution for nonlinear problems cannot be obtained easily, an alternative way is to compare a numerical solution with solutions of the other methods or the experimental data.

The comparison of numerical methods for a 2- dimensional free surface flow around a submerged hydrofoil is reported in Reference 6). The com-

putational results by the finite difference methods for the Navier-Stokes equations and by the bound-ary element methods for the potential flow theory under the same conditions are compared with each other and with the experimental data. However, the Navier-Stokes solutions are for the low Rey- nolds number flow and the comparison with the inviscid computations or the experiment at high Reynolds number is not adequate for the validation.

On the other hand, the validation of the present Navier-Stokes solver for low Reynolds number

flows is needed to develop the methods for higher Reynolds number flows. In the present paper, the finite difference method for the Euler equations with the free surface conditions are developed based on the method for the Navier-Stokes equations at low Reynolds number6) to study the basic character-istics of the numerical scheme. Because the phy-sical dissipation of energy does not exist in inviscid flows, the numerical dissipation errors of the numerical scheme are expected to appear in the Euler solution more clearly than in the Navier-Stokes solution. The examination of the Euler solution enables to estimate accuracy of the Navier-Stokes solver to some extent. The basic algo-rithms and numerical procedures of both methods are kept common as much as possible for this

purpose. The numerical procedure is described in the

following section. Several implementations of the body and free surface boundary conditions are tested by the numerical results of the Euler solver. Also, the comparison of the Euler and the Navier-Stokes solutions with the inviscid calculation and the experimental data are made for the free surface flow around NACA 0012 wing section under the same condition as a part of Reference 6).

2. Numerical scheme

2.1 Governing equations The basic equations are the continuity equation

and the Navier-Stokes/Euler equations for a unsteady incompressible fluid. They are written in the physical Cartesian coordinates (x, z t) as

( 1 )

(2 a)* Ship Research Institute

10 Journal of The Society of Naval Architects of Japan, Vol. 164

(2 b )

Where the subscripts mean the partial differential. The horizontal and vertical (upward positive) co-ordinates (x, z) are nondimensionalized by the chord length of the hydrofoil L, the (x, y) components of velocity (u,w) by the uniform flow Uo, time t by L/Uo and pressure p by pUo2. (p is the den-sity of water.) Re is Reynolds number and Fn is Froude number defined as

(3)

(4)

where v is the kinematic viscosity of water and

g is the gravitational acceleration. In the Euler equations, the diffusion terms of the right hand side of Eqs. (2) are dropped.

The body-fitted curvilinear coordinates system.

(e, C ; t) is used in the calculation. The transfor-mation from the physical coordinates to the computational ones is given as

(5)

Note that the computational coordinates are not

time-dependent, because it does not fit to the free

surface configuration which moves with time,

following the 3-dimensional scheme for the Navier-

Stokes equations developed previously') .

The equation of the continuity and the momen-

tum equations are transformed as follows.

(6)

(7 a )

(7 b )

In the Euler equations, the second terms in the right hand side are dropped again. In Eqs. (7),

(U, W) are the unscaled contravariant velocity components defined as

(8 a )

(8 b )

φ is pressure　 excluding the hydrostatic component

(9)

∇ 2 is the transformed Laplacian operator and

defined as

(10)

Andξx,ξz etc. appeared in Eqs.(6)-(10)are the

metrical quantities of the grid.

2.2 Basic algorithm

The basic algorithm of the present scheme is equal to the 3-dimensional Navier-Stokes solver of Reference 1). Velocity is updated by the finite-difference form of the momentum equations

(7) in the time-marching procedure. On each time step, pressure is calculated by the Poisson equation which is obtained from Eqs. (6) and (7) as follows.

(13)

where∇2 is defined in Eq.(10). The last term

Dt is added to avoid the accumulatiorl of numerical

errors, where D is the divergence and equal to the

left-hand-side of the Eq. (6). Dt is evaluated by・

the forward difference in which the value on the

next time step is set zero. In the Euler solver,

the diffusion terms appeared in Eq.(13)are dropp-

ed,

The time differentials in Eqs.(7)are expressed

by the forward differencing. The convection terms

are evaluated by the third-order upstream difference

which is written as, for example:

(14)

where subscripts i and k denote the grid point. The second-order central difference is used for the other spatial differentials in Eqs. (7) and (13) except for the metrical quantities of the grid for which the fourth-order central difference is used. On the boundaries the lower-order difference is used appropriately. Eq. (13) is solved iteratively by the SOR method.

2.3 Free surface conditions

The inviscid free surface conditions consist of the

following two conditions. One is the pressure

condition which means that pressure on the free

surface is equal to the atmospheric pressure. The

other is the kinematic condition which describes

that fluid particles on the free surface remain on

it.

Assume that the free surface configuration is

defined in the computational coordinates as

(15)

The pressure condition is

(16)

where Po is the atmospheric pressure. The 'irregu-

lar stars method' is used in the SOR method for

Numerical Computation of a Free Surface Flow around a Submerged Hydrofoil 11

the Poisson equation (13). The kinematic condition is written as

(17)

This condition is implemented in two different ways

and the results are compared in the following

section.

In the first way, the function H is defined as

(18)

that is, the wave height is defined on the e= constant lines. From Eqs. (17) and (18), the following condition is obtained

(19)

Eq. (19) is transformed into the finite-difference form in the same manner as the momentum equa-tions (7), that is, the forward difference in time and the third-order upstream difference in space. The velocity components (U, W) on the free surface are extrapolated equally from the velocity at the inner grid points. This treatment has the third-order accuracy but cannot cope with the overturning of waves.

The second way can handle the overturning waves by expressing the free surface configurationHin the two ways, that is,

(20a)

and

(20 b )

where wave configuration is defined as series of the

maker particles distributed on the lines of either

ξ= constaizt or ζ ＝constant. By substitution of

Eqs. (20) into Eq. (17), the equations similar to Eq. (19) are obtained and these equations are solved in the finite-difference method. The move-ment of particles on lines of =constant are calculated by the equation from Eq. (20 a) and the other particles are by that from Eq. (20 b) (see Reference 7) for detail). Accuracy of the present implementation is the first order. This method is used for comparison though the wave does not overturn in the calculations described here.

2.4 Body boundary conditions

The significant difference between the Navier-

Stokes solver and the Euler solver exists in the

body boundary condition.

The no-slip condition is used in the Navier-Stok-

es solver. that is.

on the body surface (18)

Pressure on a body is set equal to pressure at the

outer grid points adjacent to the body. When the

grid lines are orthogonal to the body, this approx-imates the pressure condition in the boundary layer

theory that the pressure gradient normal to the

wall is zero.

The Euler solver requires the free-slip condition

on the body boundary. That is, the normal

component of velocity on the body set zero.

Assumed that the body boundary is mapped into

ζ ＝ comstant line, the normal component of velocity

lS

(19)

and the tangential component is

(20)

On the body boundary, the normal component is set zero and the tangential component is linearly extrapolated9). At the trailing edge, the Kutta condition is implemented by setting the grid metrics to be the average of those of the face side and the back side in Eqs. (19) and (20). Note that the Kutta condition is not used in the computation by the finite volume method for the Euler equation by Rizzi et al.10), because no velocity is defined at the trailing edge in the staggered mesh system of the finite volume method. Pressure on the body is obtained from the normal momentum equationn)

(21)

If pressure on a body is also extrapolated, the total

pressure loss appears on the body probably because of the lack of the condition for momentum as

shown in the following section.

2.5 Other boundary conditions

On the inflow boundary, the uniform flow and

the hydrostatic pressure are given as

zl＝1.0, v＝w＝0, φ ＝0 on the inflow boundary

(19)

And the wave heigth is set zero. On the outflow boundaries, pressure, velocity

and wave height are extrapolated equally from the inside.

On the bottom boundary, w and ¢ are set zero and u is extrapolated equally from the inside. This simulates the free-slip wall on the bottom boundary, that is, the flow model is for shallow water.

3. Computed results and discussions

3.1 Computational conditions

The hydrofoil the section of which is NACA 0012 is located 1. 286 x L beneath the undisturbed free surface with the 5° angle of attack. This follows the condition of the experiment by Duncan11) and Reference 6). Froude number is 0. 567 and Reynolds number in the Navier-Stokes calculation is 10, 000, while Reynolds number at the experi-ment is about 140, 000.

The computational grid of an H-topology is shown in Fig. 1. The grid points are clustered near a body and the region that a free surface is


supposed to move in. All the computations are

carried out by this same grid for comparison,

though the more coarse grid can be used in the

Euler calculation. The number of points are 170 x

165 and the minimum spacing in c-direction near

the body is 0.0005. In the free surface region

behind the hydrofoil, the spacings in e- and ƒÄ- direc-

tion are 0. 05 and 0. 01, respectively. The region

of computation is •|4. 0•…x•…4. 0 and •|2.148•…z•…

0.20 where x=0 is the position of the mid-chord

of the hydrofoil and z=0 is the position of the

still water level.

Four calculations are carried out and the condi-

tions are tabulated in Table 1. Case-1 is the

Navier-Stokes solution with the free surface

condition for overturning waves. Case-2 is the

Euler solution in the same condition c_s Case-1 and the pressure condition on a body is to use Eq.

(21). Case-3 is the Euler solution with the free surface condition for non-overturning waves and the body condition is identical to Case-2. Case-4 is equal to Case-2 except that the linear extrapola-tion is used in the body boundary condition for

pressure. The time increment limited by the CFL condi-

tion is set 0.0002 in each calculation except that the time increment of 0. 00025 is used up to 90, 000-th time step in the Navier-Stokes computation.

The convergence of the solutions are judged by the free surface development. The Navier-Stokes solution (Case-1) did not reach the steady state because the vortex shedding occurred in the back side of the hydrofoil. The wave configura-tion, however, become almost steady and the solu-tion became periodic around 158, 000-th time step. The convergence of the Euler solutions depends on the free surface condition. Cases-2 and 4 for overturning waves reached the steady state at around 150, 000-th time step, while Case-3 with the higher-order treatment of a free surface became steady after 220, 000-th time step. The results shown below are at 160. 000-th time step for Cases-1, 2 and 4 and at 230, 000-th for Case-3.

Fig. 1 Computational grid

Table 1 Computational conditions

Fig. 2 Comparison of wave configurations


3.2 Results

The wave configurations of Cases-1, 2 and 3 are shown in Fig. 2 together with the experimental data by Duncan and the solution of the finite-differ-ence computation for the potential flow by Cole-man12). The Navier-Stokes solution (Case-1) shows the shallower wave trough above the hydrofoil than those of the other calculations or the experiment and this is the origin of the following small waves. The treatment of the free surface kinematic condition also affects the follow-ing waves us described below. In the two Euler solutions (Cases-2 and 3), the amplitude of the following waves differs from each other, though the first troughs have the almost same depth. This difference is clearly due to the free surface treatment. The first-order scheme of the free surface movement in Case-2 causes the damping of the amplitude of waves by its numerical dissipation effect in compensation for the flexibility to cope with overturning waves. The third-order scheme in Case-3 shows less damping of the wave heights and good agreement with the experimental data and the nonlinear computation by Coleman. Because more wave energy propagates to down- stream without dissipation, it takes more time steps to reach the converged solution. The wave length and the arl-iplHalo of the second crest is different

from the experiment but in good accordance with the results by Coleman. Because the calculated wave shape depends on the grid spacing and the computational domain strongly, more numerical experiments are needed to explain these differences between the computations and the experiment.

Figs. 3 show the pressure contour maps for Cases-1, 2 and 3. The two Euler solutions (Cases-2 and 3) have the almost same distribution except for the discrepancy behind the hydrofoil. This is due to the difference of the free surface treatment. The pressure contour for Case-1, the Navier-Stokes solution, shows the quite different pattern. The hydrofoil trails lines of pressure peaks in its wake which corresponds with the shed vortices and pressure change due to the following waves is least.

The velocity vector plots near the trailing edge of the hydrofoil for Cases-1, 2 and 4 are shown in Figs. 4. In the viscous solution of Case-1, the

Fig. 3 Pressure contours around the hydrofoil.

The dotted lines show negative values.

The contour interval is 0. 05 Cp.

Fig. 4 Velocity vectors around a trailing edge

of the hydrofoil.


large vortex is observed above the trailing edge. This vortex is shed into the wake periodically. In the inviscid solution that uses the momentum equations as the pressure condition on a body

(Case-2), the free-slip condition on the body and the Kutta condition at the trailing edge is satisfied seemingly. In the other inviscid solution that uses the linear extrapolation for pressure on a body

(Case-4), however, the velocity defect similar to the boundary layer appears on the back side of the hydrofoil. The implementation of the pressure condition on a body in the Euler solver affects the solution.

The same features can be seen in Figs. 5 wherethe contours of total pressure defined as (φ ＋1/2

(u2＋ ω2))/(φ ∞＋1/2　 U∞2) (φ ∞ and U∞ are static

pressure and velocity of a free stream and equal to zero and unity, respectively.) are shown. In

the inviscid flow, total pressure must be conserved. However, the two Euler solutions (Case-2, 3 and 4) show, more or less, the total presure loss on the back side and in the wake of the hydrofoil.

The magnitude of the loss near the trailing edge for the inviscid cases can be seen in Figs. 6. Case-2 and 3 with the proper body boundary condition show the loss of about 10% from the free stream value on the body surface, while Case-4 with the extrapolation of pressure on a body show about 40% loss on the body. In Case-3, total pressure loss is small and limited to the vicinity of the hydrofoil except for the region where the grid skewness is large. For the improve-ment of accuracy, together with the tuning of a computational grid, the conservative property of a numerical scheme should be considered. The

present scheme uses the non-conservative form of

Case-1

Case-2

Case-3

Ca3e-4

Fig. 5 Total pressure contours around the hydrofoil. The dotted

lines show values less than 1.0. The contour interval is

0. 05.


the convection terms and the artificial dissipation terms are added into the finite-difference form of the convection terms. These effects on the conser-vative property must be tested by the further examples.

In the viscous case (Case-1), the results in Figs. 5 show total pressure loss of large magnitude in the boundary layer and the wake. However, it should be noted that the numerical errors described in the above inviscid cases still remain and are included in the results.

Thus, the conservation of total pressure in in-viscid flows can be a measure to verify accuracy of the numerical scheme or the effect of grids by using the Euler solution. And if the same algorithm and the grid is used, the Navier-Stokes solution is expected to have errors of the same magnitude.

Fig. 7 shows the pressure distribution on the hydrofoil for the Euler solution of Case-3 together with the solution of the boundary element method for the potential flow with the nonlinear free surface condition by Suzuki6). On the face side

pressure by the Euler solution is smaller than that by the potential flow calculation. The possible reason for this difference is that the Euler solution is for the finite water depth of 2. 148 x L while the potential flow solution is for the infinite water depth. On the back side, despite of slight wig-

gles of the Euler solution around the mid-chord, agreement of the two solutions is good. It seems that a total pressure loss on the body is due to the velocity defect rather than to the pressure loss. The difference near the leading and trail-ing edges is thought to be the effect of the density

of the computational points. In Fig. 8, time history of the pressure distribu-

tion on the hydrofoil for the viscous case (Case-1) are shown together with vorticity contours. The

pressure peak on the back side is lower than that of the Euler solution, that corresponds with the shallower wave trough above the hydrofoil of the viscous solution in Fig. 2. This tendency at low Reynolds number flows are also reported by Shin et al5). The periodic behavior of the flow field due to the vortex shedding can be seen again. The period of the vortex shedding is about 0. 48 in the nondimensional time. This periodic behavi-or is also observed in the other Navier-Stokes solution for the infinite flow region at the same Reynolds number of 10, 000 by Nakamura et ali3).

The lift and drag coefficient for the viscous case (Case-1) are shown in Figs. 9. The periodic change of lift and drag in time corresponds with that of the pressure distribution in Figs. 8. The lift and drag coefficients for the Euler solutions are tabulated in Table 2. The lift and drag are also affected by the free surface and body boundary treatment.

4. Concluding remarks

1) The finite difference methods for the 2-dimensional Navier-Stokes and the Euler equations with the nonlinear free surface condition are developed.

2) The implementation of the free-slip pressure condition on a body and the free surface kinematic

Fig. 6 Total pressure contours around a trail-

ing edge of the hydrofoil. The dotted

lines show values less than 1.0. The

contour interval is 0.05.

Fig. 7 Pressure distribution on the

hydrofoil. The solid line

shows the Euler solution and

the dotted line shows the

potential flow solution.


(a) t=36.10

(b) t=36.22

(c) t=36.34

(d) t=36.46

(e) t=36.58

Fig. 8 Time history of pressure distribution and vorticity contours for the

Navier-Stokes solution. ( a ) 158, 000-th step (t =36. 10), ( b ) 158, 600-th step (t =36. 22), ( c ) 159, 200-th step (t =36. 34), ( d ) 159, 800-th step (t =36. 46) and (e) 160, 400-th step (t =36. 58)

Fig. 9 Time history of lift and drag coeffici-

ents for the Navier-Stokes solution

Table 2 Lift and drag

coefficients for

the Euler solu-

tions


condition affects the Euler solutions. The present implementation of the free surface condition for overturning waves is not accurate enough to simulate wave deformation precisely. Pressure on a body must be calculated by the momentum equation when the free-slip condition is used. The Euler solution with the proper boundary conditions is reasonable in comparison with the experiment and other inviscid calculation, though the numeri-cal errors still remain to some amount.

3) The solution by the Euler solver with the

proper boundary condition using the same algo-rithm and the grid as that by the Navier-Stokes solver can be used as a measure to estimate accuracy of the Navier-Stokes solution.

4) Some improvement of the present Navier-Stokes solver and the development of the high Reynolds number flow solver based on the informa-tion obtained here are in progress and the results will be shown in the near future.

Acknowledgement

The author would like to express his sincere

gratitude to the members of the CFD group at Ship Research Institute for their discussions and suggestions.

The calculations are carried out by HITAC M 680 at the Computer Centre of the University of Tokyo, ACOS S 910 with Scientific Attached Pro-cessor at Ship Research Institute and SUN-3/4 workstations.

References

1) Hino, T.: Numerical Simulation of a Viscous

Flow with a Free Surface around a Ship

Model, J. Soc. Nay. Archt. Jpn., Vol. 161,

pp. 1•`9 (1987)

3) Kodama, Y.: Computation of High Reynolds

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IAF Scheme, J. Soc. Nay. Archt. Jpn.,

Vol. 161, pp. 24•`33 (1987).

3) Sato, T., Miyata, H., Baba, N. and Kajitani,

H.: Finite-Difference Simulation Method for

Waves and Viscous Flows about a Ship, J.

Soc. Nay. Archt. Jpn., Vol. 160, pp. 14•`20

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6) Suzuki, K. and Hino, T.: Comparison ofCalculation Methods for 2-D Free-Surface Flow about Hydro-Foil, Proc. Open Forum on Numerical Ship Hydrodynamics, Tokyo,

pp. 107-428 (1987). (in Japanese)7) Hino, T.: Numerical Simulation Methods for

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8) Hung, C.: Extrapolation of Velocity forInviscid Solid Boundary Conditions, AIAA

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numerical computation of a free surface flow around a

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