Part II: Finite Difference/Volume Discretisation for CFD · PDF filePart II: Finite Difference/Volume Discretisation for CFD ... Module 9: Case Study Computational Fluid Dynamics,

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Part II: Finite Difference/Volume Discretisation forCFD

Finite Volume Method of the Advection-Diffusion Equation

A Finite Difference/Volume Method for the IncompressibleNavier-Stokes Equations

Marker-and-Cell Method, Staggered GridSpatial Discretisation of the Continuity EquationSpatial Discretisation of the Momentum EquationsTime DiscretisationChorin ProjectionImplementation Aspects

Tobias Neckel: Scientific Computing I

Module 9: Case Study – Computational Fluid Dynamics, Winter 2013/2014 19


Finite Volume Method –Advection-Diffusion Equation

• compute tracer concentration q with diffusion β and convection v :

−βqxx + (vq)x = 0 on Ω = (0,1)

with boundary conditions q(0) = 1 and q(1) = 0.• equidistant grid points xi = ih, grid cells [xi , xi+1]

• back to representation via conservation law (for one grid cell):∫ xi+1

xi

∂

∂xF (x) dx = F (x)

∣∣∣∣xi+1

xi

= 0

with F (x) = F (q(x)) = −βqx (x) + vq(x).• we need to compute the flux F at the boundaries of the grid cells;

however, assume q(x) piecewise constant within the grid cells




Finite Volume Method –Advection-Diffusion Equation (2)

• wanted: compute F (xi ) with F (q(x)) = −βqx (x) + vq(x)• where q(x) := qi for each Ωi = [xi , xi+1]• computing the diffusive flux is straightforward:

−βqx∣∣xi+1

= −β q(xi+1)− q(xi )

h• options for advective flux vq:

• symmetric flux:

vq∣∣xi+1

=vq(xi ) + vq(xi+1)

2• “upwind” flux:

vq∣∣xi+1

=

vq(xi ) if v > 0vq(xi+1) if v < 0




Finite Volume Method –Advection-Diffusion Equation (3)

• system of equations: for all i

F (x)

∣∣∣∣xi+1

xi

= F (xi+1)− F (xi ) = 0

• for symmetric flux:

−β q(xi+1)− 2q(xi ) + q(xi−1)

h2 + vq(xi+1)− q(xi−1)

2h= 0

leads to non-physical behaviour as soon as β < vh2

(observe signs of matrix elements!)• system of equations for upwind flux (assume v > 0):

−β q(xi+1)− 2q(xi ) + q(xi−1)

h2 + vq(xi )− q(xi−1)

h= 0

→ stable, but overly diffusive solutions (positive definite matrix)




Marker-and-Cell Method – Staggered Grid

Marker-and-Cell method (Harlow and Welch, 1965):• discretization scheme: Finite Differences• can be shown to be equivalent to Finite Volumes, however• based on a so-called staggered grid:

• Cartesian grid (rectangular grid cells),with cell centres at xi,j := (ih, jh), e.g.

• pressure located in cell centres• velocities (those in normal direction) located on cell edges




Spatial Discretisation – Continuity Equation:

• mass conservation: discretise ∇ · ~u→ evaluate derivative at cell centres, allows central derivatives:

(∇ · ~u)∣∣i,j =

∂u∂x

∣∣∣∣i,j

+∂v∂y

∣∣∣∣i,j≈ ui,j − ui−1,j

h+

vi,j − vi,j−1

h

remember: ui,j and vi,j located on cell edges• notation: (∇ · ~u)

∣∣i,j := (∇ · ~u)

∣∣xi,j

(evaluate expression at cell centre xi,j )




Spatial Discretisation – Pressure Terms

• note: velocities located on midpoints of cell edges

∂u∂t

∣∣∣∣i+ 1

2 ,j= . . . ,

∂v∂t

∣∣∣∣i,j+ 1

2

= . . .

=⇒ all derivatives need to be approximated at midpoints of celledges!

• pressure term ∇p: central differences for first derivatives(as pressure is located in cell centres)

∂p∂x

∣∣∣∣i+ 1

2 ,j≈ pi+1,j − pi,j

h,

∂p∂y

∣∣∣∣i,j+ 1

2

≈ pi,j+1 − pi,j

h




Spatial Discretisation – Diffusion Term

• for diffusion term ∆~u: use standard 5- or 7-point stencil• 2D:

∆u∣∣i,j ≈

ui−1,j + ui,j−1 − 4ui,j + ui+1,j + ui,j+1

h2

• 3D:

∆u∣∣i,j,k ≈

ui−1,j,k + ui,j−1,k + ui,j,k−1 − 6ui,j,k + ui+1,j,k + ui,j+1,k + ui,j,k+1

h2




Spatial Discretisation – Convection Terms

• treat derivatives of nonlinear terms (~u · ∇)~u:• central differences (for momentum equation in x-direction):

u∂u∂x

∣∣∣∣i+ 1

2 ,j≈ ui,j

ui+1,j − ui−1,j

2h, v

∂u∂y

∣∣∣∣i+ 1

2 ,j≈ v

∣∣xi+ 1

2 ,j

ui,j+1 − ui,j−1

2h

with v∣∣xi+ 1

2 ,j= 1

4

(vi,j + vi,j−1 + vi+1,j + vi+1,j−1

)

• upwind differences (for momentum equation in x-direction):

u∂u∂x

∣∣∣∣xi+ 1

2 ,j

≈ ui,jui,j − ui−1,j

2h, v

∂u∂y

∣∣∣∣xi+ 1

2 ,j

≈ v∣∣xi+ 1

2 ,j

ui,j − ui,j−1

2h

if ui,j > 0 and v∣∣xi+ 1

2 ,j> 0

• mix of central and upwind differences possible and used




Time Discretisation

• recall the incompressible Navier-Stokes equations:

∇ · ~u = 0∂

∂t~u + (~u · ∇)~u = −∇p +

1Re

∆~u + f

• note the role of the unknowns:→ 2 or 3 equations for velocities (x , y , and z component)

resulting from momentum conservation→ 4th equation (mass conservation) to “ close” the system;

required to determine pressure p→ however, p does not occur explicitly in mass conservation

• possible approach: Chorin’s projection method→ p acts as a variable to enforce the mass conservation as

“side condition”




Time Discretisation – Chorin Projection• continuous NSE:

ρ

(∂

∂t~u + (~u · ∇)~u

)+∇p − µ∆~u = f

∇ · ~u = 0

• semi-discrete NSE (after spatial discretisation):

A~uh + D~uh + C(~uh)~uh −MT ph = fh, (3)M~uh = 0, (4)

• derive semi-discrete continuity equation (4) w.r.t. t :

M~uh = 0. (5)

• formally invert Eq. (3) w.r.t. ~uh and insert into (5)⇒ discrete pressure Poisson equation (PPE):

(MA−1MT )︸︷︷︸=:Q

ph = MA−1(−f + D~uh + C(~uh)~uh), (6)




Time Discretisation – Chorin Projection

• updating the velocities:• discretise time with explicit method (explicit Euler, e.g.):

~u(n+1)h − ~u(n)

hτ

= . . .

• update velocity “correctly”: solve momentum equations (3)• use pressure ph of PPE solution to compute discrete

pressure gradients MT ph:

⇒ ~u(n+1)h = ~u(n)

h + τ(−D~u(n)

h − C(~u(n)h )~u(n)

h +~f (n)h + MT ph

)

• costs: solve 1 linear system of equations in each time step!




Implementation

• geometry representation as a flag field (Marker-and-Cell)

flag field as an array of booleans:0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

• input data (boundary conditions) and output data (computedresults) as arrays




Implementation (2)

Lab course “Scientific Computing – Computational Fluid Dynamics”:• modular C/C++ code• parallelization:

• simple data parallelism, domain decomposition• straightforward MPI-based parallelization (exchange of

ghost layers)• target architectures:

• parallel computers with distributed memory• clusters

• possible extensions:• free-surface flows (“the falling drop”)• multigrid solver for the pressure equation• heat transfer




Part III: The Shallow Water Equations and FiniteVolumes Revisited

The Shallow Water EquationsModelling Scenario: Tsunami Simulation

Finite Volume DiscretisationCentral and Upwind FluxesLax-Friedrichs Flux

Towards Tsunami SimulationWave Speed of TsunamisTreatment of Bathymetry Data




The Shallow Water Equations

∂

∂t

hhuhv

+

∂

∂x

huhu2 + 1

2 gh2

huv

+

∂

∂y

hvhuv

hv2 + 12 gh2

= S(t , x , y)

Comments on modelling:• generalized 2D hyperbolic PDE: q = (h,hu,hv)T

∂

∂tq +

∂

∂xF (q) +

∂

∂yG(q) = S(t , x , y)

derived from conservations laws for mass and momentum• may be derived by vertical averaging from the 3D incompressible

Navier-Stokes equations• compare to Euler equations: density ρ vs. water depth h




Modelling Scenario: Tsunami Simulation

The Ocean as “Shallow Water”??• compare horizontal (∼ 1000 km) to vertical (∼ 5 km) length scale• wave lengths large compared to water depth• vertical flow may be neglected; movement of the “entire water

column”






column”






column”






column”




Modelling Scenario: Tsunami Simulation (2)

Tsunami Modelling with the Shallow Water equations:• source term S(x , y) includes bathymetry data (i.e., elevation of

ocean floor)• Coriolis forces, friction, etc., as possible further terms• boundary conditions are difficult: coastal inundation, outflow at

domain boundariesTobias Neckel: Scientific Computing I





















Finite Volume Discretisation

• discretise system of PDEs

∂

∂tq +

∂

∂xF (q) +

∂

∂yG(q) = S(t , x , y)

• with

q :=

hhuhv

F (q) :=

huhu2 + 1

2 gh2

huv

G(q) :=

hvhuv

hv2 + 12 gh2

• basic form of numerical schemes:

Q(n+1)i,j = Q(n)

i,j −τ

h

(F (n)

i+ 12 ,j− F (n)

i− 12 ,j

)− τ

h

(G(n)

i,j+ 12−G(n)

i,j− 12

)

where F (n)i+ 1

2 ,j, G(n)

i,j+ 12, . . . approximate the flux functions F (q) and

G(q) at the grid cell boundaries




Central and Upwind Fluxes

• define fluxes F (n)i+ 1

2 ,j, G(n)

i,j+ 12, . . . via 1D numerical flux function F :

F (n)i+ 1

2= F

(Q(n)

i ,Q(n)i+1

)G(n)

j− 12

= F(Q(n)

j−1,Q(n)j

)

• central flux:

F (n)i+ 1

2= F

(Q(n)

i ,Q(n)i+1

):=

12

(F(Q(n)

i

)+ F

(Q(n)

i+1

))

leads to unstable methods for convective transport• upwind flux (here, for h-equation, F (h) = hu):

F (n)i+ 1

2= F

(h(n)

i ,h(n)i+1

):=

hu∣∣i if u

∣∣i+ 1

2> 0

hu∣∣i+1 if u

∣∣i+ 1

2< 0

stable, but includes artificial diffusion




(Local) Lax-Friedrichs Flux

• classical Lax-Friedrichs method uses as numerical flux:

F (n)i+ 1

2= F

(Q(n)

i ,Q(n)i+1

):=

12

(F(Q(n)

i

)+ F

(Q(n)

i+1

))− h

2τ(Q(n)

i+1−Q(n)i

)

• can be interpreted as central flux plus diffusion flux:

h2τ(Q(n)

i+1 −Q(n)i

)=

h2

2τ·

Q(n)i+1 −Q(n)

i

h

with diffusion coefficient h2

2τ , where c := hτ is some kind of

velocity (“one grid cell per time step”)• idea of local Lax-Friedrichs method: use the “appropriate”

velocity

F (n)i+ 1

2:=

12

(F(Q(n)

i

)+ F

(Q(n)

i+1

))−

ai+ 12

2(Q(n)

i+1 −Q(n)i

)




Wave Speed of Tsunamis

• consider the 1D case

∂

∂t

(h

hu

)+

∂

∂x

(hu

hu2 + 12 gh2

)= 0

• with q = (q1,q2)T := (h,hu)T , we obtain

∂

∂t

(q1q2

)+

∂

∂x

(q2

q22/q1 + 1

2 gq21

)= 0

• write in convective form:

∂

∂t

(q1q2

)+ f ′

∂

∂x

(q1q2

)= 0

with

f ′ =

(∂f1/∂q1 ∂f1/∂q2∂f2/∂q1 ∂f2/∂q2

)=

(0 1

−q22/q

21 + gq1 2q2/q1

)=

(0 1

−u2 + gh 2u

)




Wave Speed of Tsunamis (2)

• compute eigenvectors and eigenvalues of f ′:

λ1/2 = u ±√

gh r1/2 =

(1

u ±√

gh

)

• and then with f ′ = RΛR−1, where R := (r1, r2) andΛ := diag(λ1, λ2), we can diagonalise the PDE:

∂

∂t

(w1w2

)+ Λ

∂

∂x

(w1w2

)= 0, w = R−1q

• for small changes in h and small velocities, we thus obtain thatwaves are “advected” (i.e., travel) at speed λ1/2 ≈ ±

√gh

• recall local Lax-Friedrichs method:

F (n)i+ 1

2:=

12

(F(Q(n)

i

)+ F

(Q(n)

i+1

))−

ai+ 12

2(Q(n)

i −Q(n)i−1

)

→ choose ai+ 12

= maxλk Tobias Neckel: Scientific Computing I



Shallow Water Equations with Bathymetry

h

b

∂

∂t

hhuhv

+

∂

∂x

huhu2 + 1

2 gh2

huv

+

∂

∂y

hvhuv

hv2 + 12 gh2

=

0−(ghb)x−(ghb)y

Questions for numerics:• treat (bh)x and (bh)y as source terms or include these into flux

computations?• preserve certain properties of solutions – e.g., “lake at rest”




Shallow Water Equations with Bathymetry (2)Consider “Lake at Rest” Scenario:

• “at rest”: velocities u = 0 and v = 0• examine local Lax-Friedrichs flux in h equation:

F (n)i+ 1

2=

12

((hu)

(n)i + (hu)

(n)i+1

)−

ai+ 12

2(h(n)

i+1 − h(n)i

)= 0

⇒ F (n)i+ 1

2− F (n)

i− 12

= −ai+ 1

2

2(h(n)

i+1 − h(n)i

)+

ai− 12

2(h(n)

i − h(n)i−1

)= 0

• note: ai± 12≈√

gh and if bi−1 6= bi 6= bi+1 then hi−1 6= hi 6= hi+1

• thus: “lake at rest” not an equilibrium solution for localLax-Friedrichs flux

Additional problems:• complicated numerics close to the shore• in particular: “wetting and drying” (inundation of the coast)




References and Literature

Course material is mostly based on:• R. J. LeVeque: Finite Volume Methods for Hyperbolic Equations,

Cambridge Texts in Applied Mathematics, 2002.• M. Griebel, T. Dornseifer and T. Neunhoeffer: Numerical

Simulation in Fluid Dynamics: A Practical Introduction,SIAM Monographs on Mathematical Modeling and Computation,SIAM, 1997.

Shallow Water Code SWE:→ http://www5.in.tum.de/SWE/



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Part II: Finite Difference/Volume Discretisation for CFD · PDF filePart II: Finite Difference/Volume Discretisation for CFD ... Module 9: Case Study Computational Fluid Dynamics,