Direct Numerical Simulations of Multiphase Flows

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Numerical Methods for the Navier-Stokes Equations
Instructor Hong G. Im University of Michigan
Fall 2005
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Outline
What will be covered

• Summary of solution methods
• - Incompressible Navier-Stokes equations
• - Compressible Navier-Stokes equations
• High accuracy methods
• - Spatial accuracy improvement
• - Time integration methods

What will not be covered

• Non-finite difference approaches such as
• - Finite element methods (unstructured grid)
• - Spectral methods

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Incompressible Navier-Stokes Equations
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Incompressible Navier-Stokes Equations
The (hydrodynamic) pressure is decoupled from the
rest of the solution variables. Physically, it
is the pressure that drives the flow, but in
practice pressure is solved such that the
incompressibility condition is satisfied.
The system of ordinary differential equations
(ODEs) are changed to a system of
differential-algebraic equations (DAEs), where
an algebraic equation acts like a constraint.
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Vorticity-Stream Function Approach - 1
Vorticity-stream function formulation -
eliminating pressure
Vorticity Stream function
System of DAEs - Vorticity equation time
integration - Stream function equation
Poisson equation (SOR, etc.)
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Vorticity-Stream Function Approach - 2
Pseudo time stepping alternative to Poisson
equation
Mallinson and de Vahl Davis (1973) Back to
a system of ODEs with multiple time scales which
is integrated until convergence with an
arbitrary choice of ? that can accelerate
the convergence.
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Vorticity-Stream Function Approach - 3
Vorticity-stream function approach
Advantages - Pressure does not appear
explicitly (can be obtained later) -
Incompressibility is automatically satisfied
(by definition of stream function) Drawbacks
- Limited to 2-D applications (Revised 3-D
approaches are available)
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Vorticity-Stream Function Approach - 4
Solving pressure fields
Wall pressure at
which can be differenced as
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Vorticity-Stream Function Approach 5
Pressure fields
and using the continuity equation
Poisson equation
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Vorticity-Stream Function Approach - 6
Vorticity-Velocity Formulation A Hybrid Approach
A different (vector) form of Poisson equation
is derived from identity and using the
appropriate vector identity.
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Primitive Variable Approach
Solution Methods for Incompressible N-S Equations
in Primitive Formulation

• Artificial compressibility (Chorin, 1967)
  mostly steady
• Pressure correction approach time-accurate
• - MAC (Harlow and Welch, 1965)
• - Projection method (Chorin and Temam, 1968)
• - Fractional step method (Kim and Moin,
  1975)
• - SIMPLE, SIMPLER (Patankar, 1981)

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Artificial Compressibility - 1
Back to a system of ODE by
arbitrary constant

• With properly-chosen , solve until
• Originally developed for steady problems
• The term artificial compressibility is coined
  from
• equation of state
• Possible numerical difficulties for large

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Artificial Compressibility - 2
The concept can be applied to a time-accurate
method by using pseudo-time stepping at every
sub-steps.
At every real time step, take pseudo-time
stepping using explicit time integration until
Since the pseudo time scale is not physical, we
can accelerate the integration however we want.
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Pressure Correction Method - 1
Marker-and-Cell (MAC) Method Harlow and Welch
(1965)

• Originally derived for free surface flows with
  staggered grid

and
Explicit integration
Taking divergence of momentum equation,
Poisson equation
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Pressure Correction Method - 2
Projection Method Chorin (1968), Temam (1969)

• Originally derived on a colocated grid
• Identical to MAC except for the Poisson equation

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Pressure Correction Method - 3
MAC vs. Projection
1. Integration without pressure
2. Poisson equation
3. Projection into incompressible field
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Pressure Correction Method - 4
SIMPLE Algorithm Patankar (1981)
(Semi-Implicit Method for Pressure Linked
Equations)
- Iterative procedure with pressure correction

• Guess the pressure field
• Solve the momentum equation (implicitly)
• Solve the pressure correction equation

Exact only if
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Pressure Correction Method - 4a
Insert Derivation of Poisson Equation
Guessed equation
Converged equation
Subtracting and assuming
with
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Pressure Correction Method - 5
4. Correct the pressure and velocity 5. Go to
2. Repeat the process until the solution
converges.
Notes - Originally developed for the
staggered grid system. - The corrected
velocity field satisfies the continuity equation
even if the pressure correction is only
approximate. - Sometimes tends to be
overestimated
underrelaxation
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Pressure Correction Method - 6
SIMPLER (SIMPLE Revised) - Incorporating the
projection method (fractional step)

• Guess the velocity field
• Solve momentum equation without pressure
• Solve the pressure Poisson equation

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Pressure Correction Method - 7

• Solve the momentum equation with
  implicitly
• Pressure correction equation
• Correct the velocity, but not the pressure
• Go to 2. Repeat the process until solution is
  converged.

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Pressure Correction Method - 8
Notes on SIMPLER Algorithm - If the initial
velocity field is a correct one, then
is also the correct one, so no further
iteration is needed. - Then the difference
between SIMPLER and projection method is
simply the time integration scheme SIMPLER
implicit, iterative commonly used
in steady problems Projection explicit or
factorization (Kim Moin) well
suited for unsteady problems - Iterative
procedure can benefit from multigrid method.
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Pressure Correction Method - 9
PISO (pressure-implicit with splitting of
operators) Issa (1985) Fully Implicit
Pressure equation (take divergence of momentum
eq.)
since
( in general)
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Pressure Correction Method - 10

• PISO Algorithm
• Predictor step using (same as SIMPLE)
• First corrector step

(Implicit)
in general
(Implicit)
(derive)
(Explicit)
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Pressure Correction Method - 11
3. Second corrector step
(Implicit)
(derive)
(Explicit)
More corrections can be made, but not necessary
because splitting errors for the first and second
steps are sufficiently small.
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Accuracy Improvement Spatial 1
Spatial Accuracy

• Explicit differencing - use larger stencils

• Tridiagonal - Padé (compact) schemes

• Pentadiagonal

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Accuracy Improvement Spatial 2
Ref Kennedy, C. A. and Carpenter, M. H.,
Applied Numerical Mathematics, 14, pp. 397-433
(1994) .
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Stability Consideration
Explicit time integration in 2-D requires the
stability condition
High-Re flow advection-controlled Low-Re flow
diffusion-controlled
Use implicit schemes for appropriate terms!
at both limits!
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Accuracy Improvement Temporal 1
Temporal Accuracy

• Implicit Crank-Nicolson

Nonlinear advection term requires iteration.
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Accuracy Improvement Temporal 2
Linearization of Advection Terms

• For example, a 2-D equation

can be linearized as
where
Jacobian matrix
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Accuracy Improvement Temporal 3
Fractional Step Method Kim Moin (1985)

• Projection method extended to higher accuracy

Adams-Bashforth (AB2)
Crank-Nicolson
Note that is different from the original
pressure
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Accuracy Improvement Temporal 4
Treatment of implicit viscous terms
Factorizing,
TDMA in three directions
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Accuracy Improvement Temporal 5
Notes on Fractional Step Method

• Originally implemented into a staggered grid
  system
• Later improved with 3rd-order Runge-Kutta method
• Ref Le Moin, J. Comp. Phys., 92369 (1991)
• The method can be applied to a variable-density
  problem
• (e.g. subsonic combustion, two-phase flow)
  where
• Poisson equation becomes
• Ref Rutland, Ph. D. Thesis, Stanford
  University (1989)
• Bell, Collela and Glaz, JCP, 85257
  (1989)

Eq. of State
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Boundary Conditions
Boundary Conditions for Incompressible Flows

• In general, boundary condition treatment is
  easier
• than that for the compressible flow formulation
  due to
• the absence of acoustics
• Typical boundary conditions
• - Periodic etc.
  
• - Inflow conditions
• - Outflow conditions convective outflow
  condition

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Compressible Navier-Stokes Equations
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where
Constitutive relations
or
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Solution methods for compressible N-S equations
follows the same techniques used for hyperbolic
equations

• For smooth solutions with viscous terms, central
  differencing
• usually works.
• No need to worry about upwind method,
  flux-splitting,
• TVD, FCT (flux-corrected transport), etc.
• In general, upwind-like methods introduce
  numerical
• dissipation, hence provides stability, but
  accuracy
• becomes a concern.

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Explicit Methods
- MacCormack method - Leap frog/DuFort-Frankel
method - Lax-Wendroff method - Runge-Kutta method
Implicit Methods
- Beam-Warming scheme - Runge-Kutta method
Most methods are 2nd order. The Runge-Kutta
method can be easily tailored to higher order
method (both explicit and implicit).
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Most of the time, an implicit integration method
involves nonlinear advection terms
which are linearized as
ADI, factorization, etc.
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Ultimately, compressible Navier-Stokes equations
can be written as a system of ODEs
Initial condition
Solution techniques for a system of ODE
applies. - Explicit vs. Implicit (Nonstiff vs.
Stiff) - Multi-stage vs. Multi-step
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Boundary Conditions
Boundary Conditions for Compressible Flows

• In general, boundary condition for the
  compressible flow
• is trickier because all the acoustic waves must
  be
• properly taken care of at the boundaries.
• Typical boundary conditions
• - Periodic still easy to implement
• - Both inflow and outflow conditions require
  treatment of
• characteristic waves
• (hard-wall, nonreflecting, sponge, etc).