HOW TO SOLVE THE NAVIER-STOKES EQUATION

Benk JanosDepartment of Informatics, TU München

JASS 2007, course 2:Numerical Simulation: From Models to Software

Based on:

ON PRESSURE BOUNDARY CONDITIONS FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONPhlilp M. Gresho and Robert L. Sani

International Journal For Numerical Methods In Fluids, Vol 7, 1111-1145(1987)

Content

Short introduction Analysis of the continuum equation Pressure Poisson equation Boundary conditions Discrete approximation to the

continuum equation

Short introduction

- Important field of application of the numerical simulation- The flow is a result of different physical processes - Numerical flow simulation has a various fields of

application, real scenario simulations.

http://www.cfd-online.com/Links/misc.html#picts

Analysis of the continuum equation

gugradPugradut

u

Re

1

0udiv

The second equation is the continuity equation

The momentum equation for incompressible fluids

If it would be compressible fluid

tudiv

(1)

(2)

(3)

Analysis of the continuum equation

t

u

Each part from the (1) equation has a contribution to the momentum

The velocity change describing the acceleration of a infinite mass point. It must be in balance with …

ugradu … the convective term describing the frictionless acceleration induced by the velocity filed.

gradP … the gradient of the pressure. (by definition is an acceleration)

Analysis of the continuum equation

uRe

1 This component reflects the interior drag of the fluid.

Re is the Reynolds number.

The drag appears between two layers of fluid with different velocity.

- The friction force is acting against the velocity gradient

- This laminar flow, which opposes turbulent flow

Analysis of the continuum equation

- At gas flow this term can be neglected, but by fluids not (e.g. honey)

-The internal friction is also called viscosity, which characterize each fluid.

vs - mean fluid velocity, L - characteristic length, μ - (absolute) dynamic fluid viscosity, ν - cinematic fluid viscosity: ν = μ / ρ, ρ - fluid density.

source:http://en.wikipedia.org/wiki/Reynolds_number

Analysis of the continuum equation

We need boundary conditions so that the problem will be solvable

- Ω is the fluid domain , and is bounded by Γ

gExternal accelerations. ex: gravity

This component contains other forces which are not represented by other terms

Analysis of the continuum equation

- This leads to the BC for the continuity equation (mass conservation) n is the normal vector on the boundary

0wn (4)

w is the velocity on the boundary (Dirichlet BC) u = w(x,t) on Γ

nxwnu 0,0

We consider the initial situation t=0

Analysis of the continuum equation

In 2D we have the following vectors on the boundary

n -> normal vector

τ -> tangential component

00 uAccordingly to equation (2), this holds also for the initial velocity field in Ω

Pressure Poisson equation

Calculate the pressure field from a velocity field.

Want to have a relation between pressure and velocity, in discrete and in continuum time.

Show one way to get the Poisson equation

Pressure Poisson equation

ugradPugradut

u

Re

1

uPuutu 22

Re

1/

We start with equation (1), and we neglect external influences

First we apply the divergence operator, so we get:

We use the following expression to process further the equation:

uuu 2

Pressure Poisson equation

0rotAdiv

)/)()(Re

1( 22 tuuPuu

And because:

for any (differentiable) vector field.

We obtain:

We obtain the pressure Poisson equation:

uuuP 22

Re

1

Pressure Poisson equation

Puuut

uu

20

Re

1

A slightly modified system has to be solved for the discrete time dependent solution.

Discretizing the first(1) equation:

Expressing u+ and insert the solution in the second(2) equation, we can calculate the pressure

0 u

Boundary condition To complete the specification of the

problem for the pressure, we must set BC’s (boundary conditions) on Γ

It is important how we include the boundary conditions in our system.

For the boundary condition we need a scalar value, so we have 2 choice: either the normal or the tangential projection

Boundary condition

nnn uutuunPPn )/(Re

1/ 2

on Γ

The BC with the tangential component (Dirichlet):

First we choose the normal component: (Neumann)

uutuuPP )/(Re

1/ 2

It has been proven that these 2 conditions are equivalent.

We can calculate the pressure up to an arbitrary additive constant!!!

on Γ

Boundary condition

fuuuPtu 2

Re

1/

0/ tu

The question is: When the original NS problem is well-posed, so is the associated Poisson/Neumann problem?

Here is the proof: We start with the first equation

and in Ω

also )/(/ tufnnP on Γ

Boundary condition

fP2

fnnP /

fntufn )/(

Apply Green formula, and we obtain an integration on the boundary

Substituting the left side, we obtain the following equation

Boundary condition

),(),()/(0 txwndt

dtxwntun

Finally we obtain the following equation

w is representing the velocity on the boundary.

We obtained the mass conservation law (4), which is satisfied.

This means that our transformed problem is also well posed.

Discrete approximation to the continuum equation

Build the system of equations We want to calculate the real

pressure, on the boundary Check whether the system has a

correct boundary result

Discrete approximation to the continuum equation

We can write the momentum and the continuity equation in the following discretized general way:

)(tgDu

tfKuGPuuAuM

M – mass matrix, (for equidistant discretization is the unit matrix)

A – advection matrix

G – gradient matrix

K – diffusion matrix

D – divergence matrix

Discrete approximation to the continuum equation

TGD

uuAtfKub

)()(

f(t) , g(t) - represents the effects of the boundary Dirichlet conditions on velocity.

We use the following notations and equalities:

Denoting G with C we get:

g

b

u

P

C

CMT 0

Discrete approximation to the continuum equation

0tgPTH

We consider the hydrostatic pressure P=(1->1),

the system has a solution if g(t) satisfies the following condition.

From the above define system we can derive the discrete pressure Poisson equation:

gtubMCPCMC TT ,11

Discrete approximation to the continuum equation

We use the following staggered mesh:

We apply the continuity equation for cell with P0(on the boundary)

0

h

vv

l

uu snwe

Discrete approximation to the continuum equation

nnn uutuunPPn )/(Re

1/ 2

Expressing all 3 terms (ue is known on the boundary) from the momentum equation.

We use Taylor expansion to express the velocity at the “fictive” nodes, and we use the boundary velocities

With l, h ->0 (on the limit) we get the real boundary condition.

)()/(4

3/)32( 222 lOxvlvvv nmn

Thank you for your attention!