[PPT] - Finite Volume Method Praveen. C Computational and Theoretical Fluid PowerPoint Presentation

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Finite Volume Method

Praveen. C

Computational and Theoretical Fluid Dynamics Division National Aerospace Laboratories Bangalore 560 017 email: praveen@cfdlab.net

Workshop on Advances in Computational Fluid Flow and Heat Transfer Annamalai University October 17-18, 2005

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Topics to be covered

1. Conservation Laws
2. Finite volume method
3. Types of finite volumes
4. Flux functions
5. Spatial discretization schemes
6. Higher order schemes
7. Boundary conditions
8. Accuracy and stability
9. Computational issues
10. References

Hyperbolic equations, Compressible flow, unstructured grid schemes

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Conservation Laws and FVM

Basic laws of physics are conservation laws - mass, momentum, energy
Differential form

∂U ∂t + ∂f ∂x + ∂g ∂y + ∂h ∂z = 0 U - conserved variables f, g, h - flux vector

Compressible flows - shocks and other discontinuities
Classical solution may not exist
Integral form (using divergence theorem)

∂ ∂t

Ω

Udxdydz +

∂Ω

(fnx + gny + hnz)dS = 0 Rate of change of U in Ω = - (Net flux across the boundary of Ω) ⇓ Starting point for finite volume method

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Discontinuities are a consequence of conservation laws
Rankine-Hugoniot jump conditions [9, 10]

(fnx + gny + hnz)R − (fnx + gny + hnz)L = s(UR − UL) n U U

L R

Shock

Solution satisfying integral form - weak solution
Definition (Weak solution)
1. Satisfies the differential form in smooth regions
2. Satisfies jump condition across discontinuities
Hyperbolic conservation laws - non-uniqueness
Limit of a dissipative model: Navier-Stokes → Euler

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Entropy condition - second law of thermodynamics
Entropy satisfying weak solution - unique (Kruzkov)
Conservative scheme (FVM) - correct shock location (Warnecke)
Useful for solving equations with discontinuous coefficients
FVM can be applied on arbitrary grids - structured and unstructured

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Entropy condition - second law of thermodynamics
Entropy satisfying weak solution - unique (Kruzkov)
Conservative scheme (FVM) - correct shock location (Warnecke)
Useful for solving equations with discontinuous coefficients
FVM can be applied on arbitrary grids - structured and unstructured

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Entropy condition - second law of thermodynamics
Entropy satisfying weak solution - unique (Kruzkov)
Conservative scheme (FVM) - correct shock location (Warnecke)
Useful for solving equations with discontinuous coefficients
FVM can be applied on arbitrary grids - structured and unstructured

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FVM in 1-D

Divide computational domain [a, b] into N cells

a = x1/2 < x3/2 < . . . < xN+1/2 = b Ci = [xi−1/2, xi+1/2]

Ci h i−3/2 i−1/2 i+1/2 i+3/2 i

Conservation law for cell Ci

∂ ∂t xi+1/2

xi−1/2

Udx + f(xi+1/2, t) − f(xi−1/2, t) = 0

Cell average value

Ui(t) = 1 hi xi+1/2

xi−1/2

U(x, t)dx

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Conservation law for cell Ci

hi dUi dt + f(xi+1/2, t) − f(xi−1/2, t) = 0

U U i+1/2

i i+1

Riemann problem at each interface
Numerical flux function (Godunov approach)

Fi+1/2(t) = F(Ui(t), Ui+1(t))

Semi-discrete update equation (ODE system)

dUi dt = − 1 hi [Fi+1/2(t) − Fi−1/2(t)]

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Method of lines approach

– Discretize in space – Integrate the ODE system in time

Explicit Euler scheme [ U n

i ≈ U(xi, tn) ]

Ui(tn+1) − Ui(tn) ∆t = − 1 hi [Fi+1/2(tn) − Fi−1/2(tn)] ⇓ U n+1

i

= U n

i − ∆t

hi [F n

i+1/2 − F n i−1/2]

Conservation: Telescopic collapse of fluxes
i

hi dUi dt = −

i

[Fi+1/2(t) − Fi−1/2(t)] = −[f(b, t) − f(a, t)]

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Numerical Flux Function

Simple averaging

Fi+1/2 = f((Ui + Ui+1)/2)

r

Fi+1/2 = (fi + fi+1)/2

Equivalent to central differencing

dUi dt + 1 hi (fi+1 − fi−1) = 0 (unstable)

Two approaches
1. Central differencing with artificial dissipation [13]

Fi+1/2 = 1 2(fi + fi+1) − di+1/2

2. Upwind flux formula [9, 10, 13, 20, 22]

FVS: Fi+1/2 = f +(Ui) + f −(Ui+1) FDS: Fi+1/2 = 1 2(fi + fi+1) − 1 2[(∆f)−

i+1/2 − (∆f)+ i+1/2]

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Example: convection-diffusion equation

∂U ∂t + ∂f ∂x = 0, f = aU − ν∂U ∂x Fi+1/2 = aUi+1/2 − ν ∂U ∂x

i+1/2
Upwind definition of interfacial state

Ui+1/2 =

Ui

if a ≥ 0 Ui+1 if a < 0

Central-difference for viscous term

∂U ∂x

i+1/2

= Ui+1 − Ui xi+1 − xi

Upwind numerical flux

Fi+1/2 = 1 2(aUi + aUi+1) − |a| 2 (Ui+1 − Ui) − νUi+1 − Ui xi+1 − xi

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Significance of conservative scheme

Inviscid Burgers equation

∂U ∂t + ∂ ∂x U 2 2

= 0,

f(U) = U 2 2

Rankine-Hugoniot condition

fR − fL = s(UR − UL) = ⇒ s = 1 2(UL + UR)

Non-conservative form

∂U ∂t + U ∂U ∂x = 0

Upwind scheme (assume U ≥ 0)

U n+1

i

− U n

i

∆t + U n

i

U n

i − U n i−1

h = 0

r

U n+1

i

= U n

i − ∆t

h U n

i (U n i − U n i−1)

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Initial condition

U(x, 0) =

1

if x < 0 if x > 0

Numerical solution

U n

i = U o i =

⇒ stationary shock

Exact solution (shock speed = 1/2)

U(x, t) =

1

if x < t/2 if x > t/2

Conservation form from physical considerations

U ∂U ∂t + U ∂ ∂x U 2 2

= 0
r

∂ ∂t U 2 2

+ ∂

∂x U 3 3

= 0
Jump conditions not identical: s = 2

3

U2

L+ULUR+U2 R

UL+UR

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Higher order scheme in 1-D

Constant-in-cell representation
✂✁

✁ ✄ ✄✂☎ ☎ ✆ ✆✂✝ ✝

i−3/2 i−1/2 i+1/2 i+3/2 Ci−1 Ci+1 Ci

First order accurate

|Ui − U(xi)| = O(h) h = max

i

hi

Reconstruction - evolution - projection

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Higher order scheme in 1-D

Reconstruct the variation within a cell
✂✁

✁ ✄ ✄✂☎ ☎ ✆ ✆✂✝ ✝

i−3/2 i−1/2 i+1/2 i+3/2 Ci−1 Ci+1 Ci Left state Right state

Linear reconstruction

˜ U(x) = Ui + si(x − xi), x ∈ [xi−1/2, xi+1/2]

Biased interpolant

U L

i+1/2 = Ui + si(xi+1/2 − xi),

U R

i+1/2 = Ui+1 + si+1(xi+1/2 − xi+1),

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Flux for higher order scheme

Fi+1/2 = F(Ui, Ui+1)

Reconstruction variables
1. Conserved variables - conservative
2. Characteristic variables - better upwinding but costly
3. Primitive variables (ρ, u, p) - computationally cheap
Unsteady flows - reconstruction must preserve conservation

1 hi

Ci

˜ U(x)dx = Ui

Gradients for reconstruction: backward, forward, central difference

si,b = Ui − Ui−1 xi − xi−1 , si,f = Ui+1 − Ui xi+1 − xi , si,c = Ui+1 − Ui−1 xi+1 − xi−1

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Flux for higher order scheme

Fi+1/2 = F(U L

i+1/2, UR i+1/2)

Reconstruction variables
1. Conserved variables - conservative
2. Characteristic variables - better upwinding but costly
3. Primitive variables (ρ, u, p) - computationally cheap
Unsteady flows - reconstruction must preserve conservation

1 hi

Ci

˜ U(x)dx = Ui

Gradients for reconstruction: backward, forward, central difference

si,b = Ui − Ui−1 xi − xi−1 , si,f = Ui+1 − Ui xi+1 − xi , si,c = Ui+1 − Ui−1 xi+1 − xi−1

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Solution with discontinuity

1 1/2 i−1 i i+1

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Central-difference: Non-monotone reconstruction

1 1/2 i−1 i i+1

Limited gradients [9, 12]

si = Limiter(si,b, si,f, si,c)

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FVM in 2-D

Divide computational domain into disjoint polygonal cells, Ω = ∪iCi
Integral form for cell Ci

∂ ∂t

Ci

Udxdy +

∂Ci

(fnx + gny)dS = 0

Cell average value

Ui(t) = 1 |Ci|

Ci

U(x, y, t)dxdy, |Ci| = area of Ci

Cell connectivity: N(i) = {j : Cj and Ci share a common face}

Ci Ci

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∂Ci

(fnx + gny)dS =

j∈N(i)
Ci∩Cj

(fnx + gny)dS

Approximate flux integral by quadrature

C n C

i j ij

Ci∩Cj

(fnx + gny)dS ≈ Fij∆Sij

Semi-discrete update equation

|Ci|dUi dt = −

j∈N(i)

Fij∆Sij

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Numerical flux function

Fij = F(Ui, Uj, ˆ nij)

Properties of flux function
1. Consistency

F(U, U, ˆ n) = f(U)nx + g(U)ny

2. Conservation

F(V, U, −ˆ n) = −F(U, V, ˆ n)

3. Continuity

F(UL, UR, ˆ n) − F(U, U, ˆ n) ≤ C max (UL − U, UR − U)

Flux functions [10, 13, 20, 22]

– FVS: Steger-Warming, Van Leer, KFVS, AUSM – FDS: Godunov, Roe, Engquist-Osher

Integrate in time using a Runge-Kutta scheme [5, 12]

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Grids and Finite Volumes

Elements in 2-D
Elements in 3-D

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Boundary layers - prism and hexahedra
Cell-centered and vertex-centered scheme [5, 18, 21]
Median (dual) cell

– join centroid to mid-point of sides – well-defined for any triangulation

Centroid Mid−point

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Voronoi cell

– join circum-center to mid-point of sides – smooth area variation – not defined for obtuse triangles

Containment circle tessalation

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Median and containment-circle tessalation

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Stretched triangles - median dual and containment-circle
Containment-circle finite volume

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Turbulent flow over RAE2822 airfoil: vertex-centered scheme

Mach = 0.729, α = 2.31 deg, Re = 6.5 million

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Higher order scheme in 2-D

Bi-linear reconstruction in cell Ci

˜ U(x, y) = Ui + ai(x − xi) + bi(y − yi), (x, y) ∈ Ci

✂✁

✁ ✁ ✁ ✁ ✁ ✄ ✄ ✄ ✄ ✄ ✄✂☎ ☎ ☎ ☎ ☎ ☎

U UL

R

C C

i j

Define left/right states

U L = Ui + ai(xij − xi) + bi(yij − yi) U R = Uj + aj(xij − xj) + bj(yij − yj)

Flux for higher order scheme

Fij = F(U L, UR, ˆ nij)

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Gradient estimation using
1. Green-Gauss theorem
2. Least squares fitting
Green-Gauss theorem
Ci

∇Udxdy =

∂Ci

U ˆ ndS

Approximate surface integral by quadrature

∇Ui ≈ 1 |Ci|

face
face

U ˆ ndS

Face value

Uface = 1 2(UL + UR)

Non-uniform cells

Uface = αUL + (1 − α)UR, α ∈ (0, 1)

Accuracy can degrade for non-uniform grids [4, 6, 8, 14]

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Least-squares reconstruction [3, 5]

1 2 3 4

Uo + ao(xj − xo) + bo(yj − yo) = Uj, j = 1, 2, 3, 4

Over-determined system of equations - solve by least-squares fit

min

j

[Uj − Uo − ao(xj − xo) − bo(yj − yo)]2, wrt ao, bo ao =

j

αj(Uj − Uo), bo =

j

βj(Uj − Uo)

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Limited reconstruction

– Cell-centered: Min-max [3, 5], Venkatakrishnan [5], ENO-type [1, 14] – Vertex-centered: edge-based limiter [17]

Min-max limiter

Umin ≤ Uo + ao(xj − xo) + bo(yj − yo) ≤ Umax, j = 1, 2, 3, 4 (ao, bo) ← − (φao, φbo), φ ∈ [0, 1] – Very dissipative - smeared shocks – Performance degrades on coarse grids – Stalled convergence - limit cycle – Useful for flows with large discontinuities

Venkatakrishnan limiter

– Smooth modification of min-max limiter – Better control - depends on cell size – Better convergence properties

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Vertex-centered cell: Edge-based limiter

L R i i+1 i−1 i+2

U L = Ui + 1 2Limiter

(Ui+1 − Ui), |PiPi+1|

|PiPi−1|(Ui − Ui−1)

Using vertex-gradients

U L = Ui + 1 2Limiter

(Ui+1 − Ui), (

Pi+1 − Pi) · ∇Ui

Van-albada limiter

Limiter(a, b) = (a2 + ǫ)b + (b2 + ǫ)a a2 + b2 + 2ǫ , ǫ ≪ 1

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Higher order flux quadrature

UL

1

UR

1

UL

2

UR

2

✂✁

✁ ✁ ✁ ✁ ✁ ✄ ✄ ✄ ✄ ✄ ✄✂☎ ☎ ☎ ☎ ☎ ☎

C C

i j

Quadratic reconstruction in cell Ci

˜ U(x, y) = ˜ Ui + ai(x − xi) + bi(y − yi) + ci(x − xi)2 + di(x − xi)(y − yi) + ei(y − yi)2

2-point Gauss quadrature for flux

Fij = ω1F(U L

1 , UR 1 , ˆ

nij) + ω2F(U L

2 , UR 2 , ˆ

nij)

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Discretization of viscous flux

Viscous terms

∇ · µ∇u

Finite volume discretization
Ci

(∇ · µ∇u)dV =

∂Ci

(µ∇u · ˆ n)dS

Simple averaging

∇uij = 1 2(∇ui + ∇uj) – Odd-even decoupling on quadrilateral/hexahedral cells – Large stencil size

1-D case: ut = uxx

un+1

i

= un

i + ∆t

2h(un

i−2 − 2un i + un i+2)

Correction for decoupling problem [5]

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Green-Gauss theorem for auxiliary volume

Face−centered volume i j

Least-squares gradients

– Quadratic reconstruction: gradients and hessian [3] – Face-centered least-squares

Vertex-centered scheme

– Galerkin approximation on triangles/tetrahedra – Nearest neighbour stencil

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Turbulence models

Reynolds-average Navier-Stokes equations - need turbulence models
Differential equation based models: k − ǫ, k − ω, Spalart-Allmaras
Turbulence quantities must remain positive
Discretize using first order upwind finite volume method

Example: Spalart-Allmaras model

Ci

∇ · (˜ νu)dV ≈

j∈N(i)

[(uij · ˆ nij)+˜ νi + (uij · ˆ nij)−˜ νj]∆Sij (·)± = (·) ± |(·)| 2 , uij = 1 2(ui + uj)

Coupled or de-coupled approach
Stiffness problem - positivity preserving implicit methods

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Boundary conditions

Cell-centered approach
1. Ghost cell
2. Flux boundary condition

w g Boundary Ghost cell

Inviscid flow (slip flow - zero normal velocity)

ρg = ρw, pg = pw, ug = uw, vg = −vw

Viscous flow (noslip flow - zero velocity)

ρg = ρw, pg = pw, ug = −uw, vg = −vw

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Boundary flux depends on pressure only

F(Uw, Ug, ˆ n) = function of p only

Flux boundary condition

( F · ˆ n)wall = p[0, nx, ny, 0]⊤

1. Extrapolate pressure from interior cells
2. Solve normal momentum equation [2]
Vertex-centered approach - flux boundary condition
Boundary cell in vertex-centered scheme

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Accuracy and Stability

FVM with linear reconstruction - second order accurate on uniform and

smooth grids

On non-uniform grids =

⇒ formally first order accurate

Local truncation error not a good indicator of global error [22]
r’th order reconstruction and ng Gaussian points for flux quadrature - accu-

racy is min(r, 2ng) [19]

Semi-discrete scheme

dUi dt =

j∈N(i)

aij(Uj − Ui), aij ≥ 0

Local Extremum Diminishing (LED) property - maxima do not increase and

minima do not decrease (Jameson)

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If Ui is a local maximum =

⇒ Uj − Ui ≤ 0 dUi dt =

j∈N(i)

aij(Uj − Ui) ≤ 0 = ⇒ Ui does not increase

Fully discrete scheme

U n+1

i

= (1 − ∆t

j

aij)U n

i +

j

aijU n

j ,

∆t ≤ 1

j aij
Convex linear combination

min

j∈N(i) U n j ≤ U n+1 i

≤ max

j∈N(i) U n j

Prevents oscillations (Gibbs phenomenon) near discontinuities
Stable in maximum norm

min

j

U n

j ≤ U n+1 i

≤ max

j

U n

j

Elliptic equations - discrete maximum principle

min

j∈∂Ω Uj ≤ Ui ≤ max j∈∂Ω Uj

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Data structures and Programming

Data structure for FVM

– Coordinates of vertices – Indices of vertices forming each cell

Cell-based updating
for cell = 1 to Ncell

FluxDiv = 0 for face = 1 to Nface(cell) cellNeighbour = CellNeighbour(cell, face) flux = NumFlux(cell, cellNeighbour) FluxDiv += flux end Unew(cell) = Uold(cell) - dt*FluxDiv end

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Face-based updating
FluxDiv(:) = 0

for face = 1 to Nface LeftCell = FaceCell(face,1) RightCell = FaceCell(face,2) flux = NumFlux(LeftCell, RightCell) FluxDiv(LeftCell) += flux FluxDiv(RightCell) -= flux end Unew(:) = Uold(:) - dt*FluxDiv(:)

Flux computations reduced by half - speed-up of two
Other geometric quantities - cell centroids, face areas, face normals, face

centroids

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References [1] Abgrall R., “On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation”, J. Comp. Phys., Vol. 114, pp. 45-58, 1994. [2] Balakrishnan N. and Fernandez, G., “Wall Boundary Conditions for Inviscid Compressible Flows on Unstructured Meshes”, Int. Jl. for Num. Methods in Fluids, 28:1481-1501, 1998. [3] Barth T. J., “Aspects of unstructured grids and solvers for the Euler and NS equations”, Von Karman Institute Lecture Series, AGARD Publ. R-787, 1992. [4] Barth T. J., “Recent developments in high order k-exact reconstruction on unstructured meshes”, AIAA Paper 93-0668, 1993. [5] Blazek J., Computational Fluid Dynamics: Principles and Applications, El- sevier, 2004.

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[6] Delanaye M., Geuzaine Ph. and Essers J. A., “Development and application

f quadratic reconstruction schemes for compressible flows on unstructured

adaptive grids”, AIAA-97-2120, 1997. [7] Feistauer M., Felcman J. and Straskraba I., Mathematical and Computa- tional Methods for Compressible Flow, Clarendon Press, Oxford, 2003. [8] Frink N. T., “Upwind scheme for solving the Euler equations on unstructured tetrahedral meshes”, AIAA J. 30(1), 70, 1992. [9] Godlewski E. and Raviart P.-A., Hyperbolic Systems of Conservation Laws, Paris, Ellipses, 1991. [10] Godlewski E. and Raviart P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996. [11] Carl Ollivier-Gooch and Michael Van Altena, “A high-order-accurate unstruc- tured mesh finite-volume scheme for the advection-diffusion equation”, JCP, 181, 729-752, 2002. [12] Hirsch Ch., Numerical Computation of Internal and External Flows, Vol. 1, Wiley, 1988.

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[13] Hirsch Ch., Numerical Computation of Internal and External Flows, Vol. 2, Wiley, 1989. [14] Jawahar and Kamath H., “A High-Resolution Procedure for Euler and Navier- Stokes Computations on Unstructured Grids“, Journal of Computational Physics, Vol. 164, No. 1, pp. 165-203, 2000 [15] Kallinderis Y. and Ahn H. T., “Incompressible Navier-Stokes method with general hybrid meshes”, J. Comp. Phy., Vol. 210, pp. 75-108, 2005. [16] LeVeque R. J., Finite Volume Methods for Hyperbolic Equations, Cambridge University Press, 2002. [17] Lohner R., Applied CFD Techniques: An Introduction based on FEM, Wiley. [18] Mavriplis D. J., “Unstructured grid techniques”, Ann. Rev. Fluid Mech., 29:473-514, 1997. [19] Sonar Th., “Finite volume approximations on unstructured grids”, VKI Lec- ture Notes.

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Praveen. C, CTFD Division, NAL, Bangalore

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[20] Toro E., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer. [21] Venkatakrishnan V, “Perspective on unstructured grid flow solvers”, AIAA Journal, vol. 34, no. 3, 1996. [22] Wesseling P., Principles of Computational Fluid Dynamics, Springer, 2001.