HOW TO SOLVE THE NAVIER-STOKES EQUATION

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HOW TO SOLVE THE NAVIER-STOKES EQUATION
Based on ON PRESSURE BOUNDARY CONDITIONS FOR
THE INCOMPRESSIBLE NAVIER-STOKES EQUATION Phlilp
M. Gresho and Robert L. Sani International
Journal For Numerical Methods In Fluids, Vol 7,
1111-1145(1987)
Benk Janos Department of Informatics, TU
München JASS 2007, course 2 Numerical
Simulation From Models to Software
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Content

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

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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.htmlpicts
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Analysis of the continuum equation
The momentum equation for incompressible fluids
(1)
The second equation is the continuity equation
(2)
If it would be compressible fluid
(3)
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Analysis of the continuum equation
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
the convective term describing the frictionless
acceleration induced by the velocity filed.
the gradient of the pressure. (by definition is
an acceleration)
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Analysis of the continuum equation
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
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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,
? - cinematic fluid viscosity ? µ / ?, ? -
fluid density.
sourcehttp//en.wikipedia.org/wiki/Reynolds_numbe
r
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Analysis of the continuum equation
External accelerations. ex gravity This
component contains other forces which are not
represented by other terms
We need boundary conditions so that the problem
will be solvable - O is the fluid domain , and is
bounded by G
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Analysis of the continuum equation
w is the velocity on the boundary (Dirichlet BC)
u w(x,t) on G
- This leads to the BC for the continuity
equation (mass conservation) n is the normal
vector on the boundary
(4)
We consider the initial situation t0
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Analysis of the continuum equation
Accordingly to equation (2), this holds also for
the initial velocity field in O
In 2D we have the following vectors on the
boundary n -gt normal vector t -gt tangential
component
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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

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Pressure Poisson equation
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
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Pressure Poisson equation
And because
for any (differentiable) vector field. We obtain
We obtain the pressure Poisson equation
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Pressure Poisson equation
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
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Boundary condition

• To complete the specification of the problem for
  the pressure, we must set BCs (boundary
  conditions) on G
• 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

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Boundary condition
First we choose the normal component (Neumann)
on G
The BC with the tangential component (Dirichlet)
on G
It has been proven that these 2 conditions are
equivalent. We can calculate the pressure up to
an arbitrary additive constant!!!
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Boundary condition
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 O
also
on G
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Boundary condition
Apply Green formula, and we obtain an integration
on the boundary
Substituting the left side, we obtain the
following equation
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Boundary condition
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.
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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

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Discrete approximation to the continuum equation
We can write the momentum and the continuity
equation in the following discretized general way
M mass matrix, (for equidistant discretization
is the unit matrix) A advection matrix G
gradient matrix K diffusion matrix D
divergence matrix
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Discrete approximation to the continuum equation
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
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Discrete approximation to the continuum equation
We consider the hydrostatic pressure
P(1-gt1), 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
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Discrete approximation to the continuum equation
We use the following staggered mesh We apply the
continuity equation for cell with P0(on the
boundary)
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Discrete approximation to the continuum equation
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 -gt0 (on the limit) we get the real
boundary condition.
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Thank you for your attention!