A high order FV/FE projection method for compressible low-Mach number flows

18 May 2017 A. Bermúdez*§ , S. Busto*, J.L. Ferrín*§ , E.F. Toro**, M.E. Vázquez-Cendón*§ * Departamento de Matemática Aplicada, Universidade de Santiago de Compostela § Instituto Tecnológico de Matemática Industrial (ITMATI) ** Laboratory of Applied Mathematics, DICAM, Università di Trento

Outline

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2

Mathematical model Incompressible flows Low Mach number flows Numerical Method Numerical discretization Overall method A dual mesh Transport diffusion stage Finite volume discretization Numerical flux Pressure term

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4

Viscous terms Pre-projection stage Projection stage Post-projection stage Numerical results Incompressible test The importance of density Second compressible test Ongoing research and conclusions Ongoing research Conclusions

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Incompressible flows

Incompressible flows

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Incompressible flows

Incompressible Navier-Stokes equations div ρ u = 0, ∂ρ u + div Fiwu (wu , u(x , t)) + ∇ π − div τ = fu , ∂t
τ = µ ∇ u + ∇ uT .

Time: t. Density: ρ ∈ R. Velocity vector: u = u(x , y , z, t). Conservative velocity vector: wu = ρ u Pressure: π = π(x , y , z, t).

Viscous term of the stress tensor: τ . Dynamic viscosity: µ. Source term: f u . Flux: Fiwu (wu , u(x , t)) = ui wu , i = 1, 2, 3.

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Incompressible flows

Turbulence Reynolds-averaged viscous stress tensor k − ε standard τ = τu + τ R ,


τu = µ ∇ u + ∇ uT ,
2 τ R = µt ∇ u + ∇ uT − ρkI. 3

µt = ρCµ Reynolds stress tensor, τ R . Turbulent viscosity, µt . Turbulent kinetic energy, k.

k2 , ε

Cµ = 0.09. Conservative turbulent kinetic energy, wk . Dissipation rate, ε. Conservative dissipation rate, wε .

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Incompressible flows

K − ε standard µt = ρCµ

k2 , ε

   µt ∂ρk + div F wk (wk , u(x , t)) − div µ + ∇ k + ρε = Gk + fk , ∂t σk    ∂ρε µt ε2 ε + div F wε (wε , u(x , t)) − div µ + ∇ ε + ρC2ε = C1ε Gk + fε , ∂t σε k k 2 3  µt X ∂ui ∂uj Gk ≈ + , 2 i,j=1 ∂xj ∂xi C1ε = 1.44,

C2ε = 1.92,

Cµ = 0.09,

σk = 1.0,

σε = 1.3.

Flux: F wk (wk , u(x , t)) = uwk , F wε (wε , u(x , t)) = uwε .

Source MMS terms: fk , fε . Prandtl numbers: σk , σε .

Turbulent production: Gk .

Closure constants: C1,ε , C2,ε .

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Incompressible flows

Species transport and energy Species conservation equations ∂ρ y + div F wy (wy , u(x , t)) − div ∂t

   µt ρD + ∇ y = fy . Sct

Energy conservation equation ∂ρh + div F wh (wh , u(x , t)) − div ∂t Species: y = (y1 , . . . , yNe ). Number of species: Ne . Conservative species: wy = ρ y. Enthalpy: h. Conservative enthalpy: wh = ρh.

   µt ρD + ∇ h = − div qr +fh . Sct Flux: F wy (wy , u(x , t)) = uwy , F wh (wh , u(x , t)) = uwh . Mass diffusivity coefficient: D. Schmidt number: Sct . Heat flux: qr . Source terms: f y , fh .

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Incompressible flows

Conservative variables We introduce the conservative variables     ρy1 wy1      ..   ..     .   .      ρu      1     ρy w   Ne   yNe  , ˆ = wu = ρu2  , w =        ρh   wh      ρu3      ρk   wk      ρε wε

  wu w =  . ˆ w

Then, the flux can be expressed as F wu = (F1wu |F2wu |F3wu )3×3 ,

F = (F1 |F2 |F3 )(3+Ne +1+2)×3 ,

Fiwu = ui wu =

  Fiwu   Fi =   wi ˆ w ρ

3+Ne +1+2 A FV/FE projection method for compressible low-Mach number flows

wi wu ρ

= ui w,

i = 1, 2, 3.

SHARK-FV, 2017

Mathematical model

Incompressible flows

Hence, the system of equations can be rewritten: div wu = 0,  
2 ∂ wu 1 wu T + div F (wu , u) + ∇ π − div (µ + µt ) ∇ wu + ∇ wu − wk I = fu , ∂t ρ 3     µt 1 ∂ wy + div F wy (wy , u) − div ρD + ∇ wy = f y, ∂t Sct ρ    ∂wh µt wh + div F wh (wh , u) − div ρD + ∇ = − div qr +fh , ∂t Sct ρ    µt wk ∂wk + div F wk (wk , u) − div µ + ∇ + wε = Gk + fk , ∂t σk ρ    ∂wε µt wε w2 wε + div F wε (wε , u) − div µ + ∇ + ρC2ε ε = C1ε Gk + fε , ∂t σε ρ wk wk 2 3  wk2 µt X ∂ui ∂uj µt = Cµ . , Gk = + wε 2 i,j=1 ∂xj ∂xi

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

Laminar low Mach number flows

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

Low Mach number equations

We will assume that the Mach number, M, is small enough for splitting the pressure into a spatially constant function, π ¯ , and a small perturbation, π:

p(x , y , z, t) = π ¯ (t) + π(x , y , z, t),

π = O(M −2 ) π ¯

(1)

with π ¯ (t) provided, π neglected in the state equation and retained in the momentum equation.

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

Compressible Navier-Stokes equations ∂ρ + div ρ u = 0, ∂t ∂ρ u + ∇ π − div τ + div (ρ u ⊗ u) = fu , ∂t
2 τ = µ ∇ u + ∇ uT − µ div u I, 3 π ¯ = ρRθ,

R=R

Density: ρ = ρ(x , y , z, t). Gas constant: R. Universal constant: R.

Ne X Yi , M i i=1

R = 8314 J/ (kmolK).

Molecular mass (species Yi ): Mi . Temperature: θ.

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

∂ρ + div ρ u = 0, ∂t

(2)

∂ρ u ∂F1 (u, ρ) F2 (u, ρ) F3 (u, ρ) + + + + ∇ π − div τ = fu , ∂t ∂x ∂y ∂z | {z } div(F(u,ρ))

π ¯ = ρRθ.

(3)

From (2) and (3) we get the following divergence condition:   ∂ρ ∂ π ¯ div (ρ u) = − =− ∂t ∂t Rθ so that, the system of equations to be solved reads ∂ρ u + div (F (u, ρ)) + ∇ π − div τ = f u , ∂t div ρ u = q, where

∂ q := − ∂t



π ¯ Rθ

 .

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

Species transport and energy Species conservation equations ∂ρ y + div F y ((u, y), ρ) − div (ρD ∇ y) = 0. ∂t

(4)

Energy conservation equation ∂ρh + div F h ((u, h), ρ) − div (ρD ∇ h) = 0. ∂t

(5)

Taking into account the mass conservation equation and the chain rule, equations (4) and (5) are rewritten as ∂y 1 + div (y v) − y div v − div (ρD ∇ y) = 0, ∂t ρ ∂h 1 + div (h v) − h div v − div (ρDi ∇ h) = 0. ∂t ρ A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

The resolution of the Navier-Stokes equations and the species transport and energy conservation equations are coupled by state equation,   π ¯ ∂ . π ¯ = ρRθ q=− ∂t Rθ

The following equation relates the internal enthalpy for a perfect gas and its temperature: Z θ h(θ) = hθ0 + cπ (r )dr . θ0

Standard enthalpy formation: hθ0 . Temperature of formation: θ0 . Specific heat at constant pressure: cπ .

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

Conservative variables The vector of unknowns of the new system reads       y1 wy1     ρu  ..   ..   1    .   .    , ˆ = wu = ρu2  , w  =    yNe  wyN  e    ρu3 h wh

  wu w =  . ˆ w

Then, the flux can be expressed as F wu = (F1wu |F2wu |F3wu )3×3 ,

F = (F1 |F2 |F3 )(3+Ne +1+2)×3 ,

Fiwu = ui wu =

  Fiwu   Fi =   wi ˆ ρw

wi wu ρ

= ui w,

i = 1, 2, 3.

3+Ne +1+2

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

Hence, the system of equations can be rewritten:   ∂ π ¯ div wu = q, q=− , ∂t Rθ "    T ! 1 1 ∂ wu wu + div F (wu , ρ) + ∇ π − div µ ∇ wu + ∇ wu ∂t ρ ρ   2 1 − µ div wu I = 0, 3 ρ ∂ wy 1 + div F wy (w, ρ) − wy div u − div (ρD ∇ wy ) = 0, ∂t ρ 1 ∂wh + div F wh (w, ρ) − wh div u − div (ρDi ∇ wh ) = 0, ∂t ρ Z θ h(θ) = hθ0 + cπ (r )dr . θ0

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Mathematical model

Low Mach number flows

Equations dependency Incompressible flow with turbulence: Species transport (Y)

Navier-Stokes equations (π, w)

Turbulence model (k, ε)

Energy conservation (h)

Laminar low Mach number flow: Species transport (Y) Temperature (θ)

Navier-Stokes equations (π, w) Energy conservation (h)

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

State equation

Numerical Method

Numerical discretization

Numerical Method

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Numerical discretization

Numerical discretization Let Wn , π n be an approximation of the conservative variables, w (x , y , z, t n ), and of the pressure perturbation, π (x , y , z, t n ). Then Wn+1 , π n+1 can be defined from the following system of equations: n+1

f W u

− Wnu + div (F wu (Wn , ρn )) + ∇ π n − div (τ n ) = f nu , ∆t

(6)

n+1

f Wn+1 −W u u ∆t

" n

τ =µ ∇



1 n W ρ u



 +∇

1 n W ρ u


+ ∇ π n+1 − π n = 0,

(7)

div Wn+1 = q n+1 , u

(8)

T

2 − div 3

A FV/FE projection method for compressible low-Mach number flows



1 n W I ρ u

# ,

SHARK-FV, 2017

Numerical Method

Numerical discretization

n+1

f W y

   − Wny 1 n + div F wy (Wn , ρ) − div ρD ∇ Wy = 0, ∆t ρ

(9)

n+1

f Wn+1 −W y y = f y, ∆t   f n+1 − W n W Whn n wh h h + div F (W , ρ) − div ρD ∇ = − div Qnr , ∆t ρ

hn+1

f n+1 Whn+1 − W h = fh , ∆t Z θn+1 cπ (r )dr , = hθ0 +

(10)

(11)

(12)

(13)

θ0

ρn+1 = Rθn+1

π . Ne X Y n+1 i

i=1 A FV/FE projection method for compressible low-Mach number flows

Mi

SHARK-FV, 2017

(14)

Numerical Method

Overall method

Overall method Transport-difusion stage: Equations (6), (9) and (11) are solved by a FVM. Species and energy variables are updated from equations (10) and (12). Pre-projection stage: temperature and density are computed from the updated species and enthalpy using (13) and (14). Projection stage: a FEM is applied to (7)-(8) in order to obtain the pressure correction. n+1

f Post-projection stage: W is updated by using δ n+1 , (7) (the new u approximation, Wn+1 , satisfies the divergence condition). u

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

A dual mesh

A dual finite volume mesh Barycenter of the faces of the tetrahedra: {Ni , i = 1, . . . , M}. Set of the neighbours of Ni : Ki . Volume of Ci : vol(Ci ). Boundary ofSCi : Γi = ∂Ci = j∈Ki Γij . Outward normal to Γij : ηij , (ηij = η˜ij ∗ kηij k). P P Area of Γi : Si = j∈Ki S(Γij ) = j∈Ki kηij k.   Γ = Γ T Ω, ib i At the boundary (Ni ∈ Γ):  η outward normal. ib

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Transport diffusion stage

Transport-diffusion stage

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Finite volume discretization

Finite volume discretization Integrating equations (6), (9) and (11), over the finite volume Ci and applying Gauss theorem we obtain  vol (Ci ) fn+1 Wu i − Wnu i + ∆t

Z F

wu

(Wnu , ρn ) η˜dS

Z Ci

Z +

n

Z

F wY (Wn , ρ) η˜dS −

Γi

 vol (Ci )  f n+1 Whi − Whin + ∆t

Z F Γi

wh

Z

f nu dV ,

(τ ) η˜dS + Γi

 vol (Ci ) fn+1 WY i −WnY i + ∆t

∇ π n dV

=−

Γi

Ci

 Z  Z Wn ρD ∇ Y η˜dS = fY dV , ρ Γi Ci

 Whn ρD ∇ η˜dS (W , ρ) η˜dS − ρ Γ Z i Z =− qrn η˜dS + fh dV . n

Z 

Γi A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Ci

Numerical Method

Numerical flux

Numerical flux Γi is split into the cell interfaces, Γij , Nj ∈ Ki , Z Z X Z div [F (Wn , ρn )] dW = F (Wn , ρn ) η ˜ dS = F (Wn , ρn ) η ˜ ij dS Ci

Γi

Nj ∈Ki

Γij

Numerical flux on a cell boundary, Γi Z (Wn , ρn , η ˜ ) := F (Wn , ρn ) η ˜. The integral on Γij is approximated by an upwind scheme, Z

F (Wn , ρn ) η ˜ ij dS ≈ φ (Wni , ρni ) , Wnj , ρnj , η ij , Γij

with φ the numerical flux. A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Numerical flux

Numerical Flux: Rusanov scheme

1


φ (Wni , ρni ) , Wnj , ρnj , η ij = Z n Wni , η ij + Z n Wnj , η ij 2

1 n n − αRS (Wi , ρi ), (Wjn , ρnj ), η ij Wnj − Wni . 2 Coupled
 αRS (Win , ρni ), (Wjn , ρnj ), η ij = max 2 Ui ·η ij , 2 Uj ·η ij . Decoupled    wu αRS (Wnui , ρni ), (Wnuj , ρnj ), η ij = max 2 Ui ·η ij , 2 Uj ·η ij ,    cn , ρn ), (W cn , ρn ), η = max Ui ·η , Uj ·η . α ˆ RS (W ij ij ij i i j j

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Numerical flux

We consider two different schemes to obtain second order: CVC Kolgan-type scheme A. Bermúdez, S. Busto, M. Cobas, J.L. Ferrín, L.Saavedra and M.E. Vázquez-Cendón. “Paths from mathematical problem to technology transfer related with finite volume methods”. Proceedings of the XXIV CEDYA / XIV CAM (2015). Local ADER scheme S. Busto, E.F. Toro and M.E. Vázquez-Cendón. “Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations”. J. Comp. Phys. Volume 327, 15 December, Pages 553–575(2016).

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Numerical flux

A new Kolgan-type scheme: CVC ∂f (W ) ∂W (x , t) + (x , t) = 0 ∂t ∂x

New upwind second order scheme
∆t n n n n n+1 n φ Wj , Wj+1 , WL,j+1/2 , WR,j+1/2 Wj = Wj − ∆x n

n

n

n

−φ Wj−1 , Wj , WL,j−1/2 , WR,j−1/2





Numerical flux φ (U, V , UL , VR ) =

f (U) + f (V ) 1 − |Q (UL , VR )| (VR − UL ) 2 2 L

∗ WL,j−1/2 = Wj−1 + ∆j−1 , L

WL,j+1/2 = Wj + ∆j ∗ ,

R

WR,j−1/2 = Wj + ∆j ∗ , R

∗ WR,j+1/2 = Wj+1 + ∆j+1 .

∆Lj ∗ , ∆Rj ∗ the left and right limited slopes at the node xj :

 L ∆ ∗ = j

min

 R ∆ ∗ = j

max

 0, min  0, max

 max 0, min  min

0, max

1 2 1 2 −1 2 −1 2

Wj − Wj−1 Wj − Wj−1



 ,W −W
j+1 j
 , Wj+1 − Wj



 Wj+1 − Wj , Wj−1 − Wj

Wj+1 − Wj , Wj−1 − Wj

Wj+1 − Wj > 0 Wj+1 − Wj < 0 Wj−1 −Wj > 0 Wj−1 −Wj < 0

L. Cea and M.E. Vázquez-Cendón. “Analysis of a new Kolgan-type scheme motivated by the shallow water equations”. Appl. Num. Math. 62 489–506 (2012). A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

n WijL = Win + ∆ijL ,

Numerical flux

n WijR = Wjn − ∆ijR ,

∆ijL and ∆ijR are the left and right limited slopes at the face defined with the Galerkin gradients computed at the upwind tetrahedra TijL and TijL , and taking into account some limiter:

∆ijL = Lim 21 ∇Wn |TijL · Ni Nj , Wnj − Wni ,

∆ijR = Lim 12 ∇Wn |TijR · Ni Nj , Wnj − Wni , the numerical flux reads
n n φ (Win , ρni ), (Wjn , ρnj ), (WijL , ρnijL ), (WijR , ρnijR ), η ij
1 = Z(Win , ρi , η ij ) + Z(Wjn , ρnj , η ij ) 2

1 n n n n − αRS (WijL , ρnijL ), (WijR , ρnijR ), η ij WijR − WijL . 2

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Numerical flux

Local ADER Step 1. ENO-based data reconstruction1 . First-degree polynomials of the conservative variable are defined in each cell at the neighbouring of the faces: i

pi Nij (N) = Wi + (N − Ni ) (∇W )ij , j

pj Nij (N) = Wj + (N − Nj ) (∇W )ij .   (∇W )TijL (∇W )TijL ·∆N ≤ (∇W )Tij ·∆N , i (∇W )ij =  (∇W ) (∇W ) ·∆N > (∇W ) ·∆N ; Tij TijL Tij   (∇W )TijR (∇W )TijR ·∆N ≤ (∇W )Tij ·∆N , j (∇W )ij =  (∇W ) (∇W ) ·∆N > (∇W ) ·∆N , Tij TijR Tij ∆N := Nij − Nj . 1 E.F. Toro "Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Third edition". Springer, 2009. A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Numerical flux

Local ADER Step 1. ENO-based data reconstruction1 . First-degree polynomials of the conservative variable are defined in each cell at the neighbouring of the faces: i

pi Nij (N) = Wi + (N − Ni ) (∇W )ij , j

pj Nij (N) = Wj + (N − Nj ) (∇W )ij . Step 2. Computation of boundary extrapolated values at the barycentre of the face Γij , Nij . i

Wi Nij = pi Nij (Nij ) = Wi + (Nij − Ni ) (∇W )ij , j

Wj Nij = pj Nij (Nij ) = Wj + (Nij − Nj ) (∇W )ij .

1 E.F. Toro "Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Third edition". Springer, 2009. A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Numerical flux

Step 3. Taylor series expansion in time and Cauchy - Kovalevskaya procedure are applied to locally approximate the conservative variables at time τ = ∆t : 2 Wi Nij = Wi Nij − +


∆t αiNij ∇W|TijL η ij + αjNij ∇W|TijR η ij , 2L2ij

Wj Nij = Wj Nij − + Lij = min

n


∆t Z(Wi , η ij ) + Z(Wj , η ij ) 2Lij


∆t Z(Wi , η ij ) + Z(Wj , η ij ) 2Lij


∆t αiNij ∇W|TijL η ij + αjNij ∇W|TijR η ij , 2L2ij

vol(Ci ) vol(Cj ) , S(Cj ) S(Ci )

o

, αi Nij diffusion coefficient.

Step 4. Computation of the numerical flux considering Rusanov scheme: φ



 



WinNij , ρni , WjnNij , ρnj , ηij



=

1 Z WinNij , ηij 2

 

1 − αRS (WinNij , ρni ), (WjnNij , ρnj ), η ij 2



A FV/FE projection method for compressible low-Mach number flows





+ Z WjnNij ηij



 

WjnNij − WinNij .

SHARK-FV, 2017

Numerical Method

Pressure term

Pressure term Pressure term Z

n

∇ π dV = Ci

X Z Nj ∈Ki

π n η˜ij dS.

Γij

The pressure is approximated by the average of the values at the vertices of the face. Z

1 n (π (V1 ) + π n (V2 ) + π n (B)) area (Γij ) η˜ij = 3 Γij   5 1 n n n n = (π (V1 ) + π (V2 )) + (π (V3 ) + π (V4 )) ηij . 12 12 π n η˜ij dS ≈

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Viscous terms

Viscous terms Viscous term of the momentum equation Z

n

div τ dV = Ci

X Z Nj ∈Ki

Γij

n

τ η˜ij dS =

X Z Nj ∈Ki

µ ∇ u η˜ij dS.

Γij

Gradient approximation: 1

orthogonal and non-orthogonal flux in the face 2 ;

2

Galerkin approximation at the tetrahedron of the face;

3

average of the Galerkin approximation at the upwind tetrahedra;

4

average of the Galerkin approximation at the three tetrahedra.

2 A. Bermúdez, J.L.Ferrín, L. Saavedra and M.E. Vázquez-Cendón "A projection hybrid finite volume/ element method for low-Mach number flows". J. Comp. Phys. 271 360–378, 2014. A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Viscous terms

Galerkin approximation at the tetrahedron of the face: We consider a numerical diffusion function such that Z
µ∇Un η˜ij dS ≈ ψu Uni , Unj , η ij . Γij

Two different discretizations are considering depending on the convection term treatment: CVC Kolgan type scheme
ψu Uni , Unj , η ij = µ (∇Un )|Tij η ij . Local ADER scheme

ψu Uni , Unj , η ij = µ ∇Un |T η ij , ij   ∆t Uni = Uni + tr ∇Vn|TijL + ∇Vn|Tij , 4  1 n n n ∇U|T + ∇U Vi = µ |Tij . ijL 2 A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Viscous terms

c Viscous terms for W The viscous terms for the species and energy equations read Z   X Z 1 cn dV = 1 cn η div D n ∇W D n ∇W ˜ ij dS. ρ Ci ρ Γij

(15)

Nj ∈Ki

Like for the momentum equation we introduce a new numerical diffusion function, ψwˆ , such that Z   cn η cn , W cn , η , D n ∇W ˜ dS ≈ ψb W ij i j w Γij

    cn , W c n , η = D n ∇W cn ψb W η ij . ij i j ij w |Tij

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Pre-projection stage

Pre-projection stage: Low Mach number flows

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Pre-projection stage

Pre-projection stage. Low Mach number flows The temperature is calculated by solving hn+1 = hθ0 +

Z

θ n+1

cπ (r )dr θ0

using Newton’s method. The density is obtained from the state equation. The computed density is used to approximate the source term of the projection stage equations: q n+1 =

ρn+1 − ρn . ∆t

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Projection stage

Projection stage

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Projection stage

Projection stage We use a finite element method in order to solve n+1

f Wn+1 −W u u ∆t


+ ∇ π n+1 − π n = 0, div Wn+1 = q n+1 . u

 R Let z ∈ V0 be a test function, V0 := z ∈ H 1 (Ω) : Ω z = 0 , then Z Z Z
n+1 1 1 n+1 n f ∇ π − π · ∇zdV = W · ∇zdV − Wn+1 ·∇zdV . ∆t Ω u ∆t Ω u Ω

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Projection stage

Using the divergence condition and the Green formula, we obtain Z Z Z Z Z
n+1 n+1 n+1 Wn+1 ·∇zdV = W ·η∇zdS− div W zdV = g z− q n+1 zdS. u u u Ω

Γ

Ω

Γ

Ω

Replacing the previous expression in the variational formulation and introducing the variable δ n+1 := π n+1 − π n , to obtain π n+1 we need to solve the weak problem Z Z Z Z n+1 1 1 1 n+1 f W · ∇zdV + q zdV − g n+1 zdS. ∇δ n+1 · ∇zdV = u ∆t ∆t ∆t Ω Ω Γ Ω This weak problem can be seen as corresponding to the following Laplace problem with Neumann conditions  n+1 1  n+1 f div W − q in Ω, u ∆t  1 fn+1 Wu · η − g n+1 = in Γ. ∆t

∆δ n+1 = ∂δ n+1 ∂η

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical Method

Post-projection stage

Post-projection stage

Linear momentum density n+1

n+1 f Wn+1 − πin u i = Wu i + ∆t∇ πi

A FV/FE projection method for compressible low-Mach number flows




SHARK-FV, 2017

Numerical results

Numerical results

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

Incompressible test

Incompressible test Computational domain:  Ω = (x , y , z) ∈ R3 | (x 2 + y 2 ) ≤ 0.52 , z ∈ [0, 5] . Fluid properties: ρ = 0.38474 µ = 1.7894e −5 Boundary conditions:   2 −y 2 ΓI1 : v = 0, 0, −x0.25 +1 , ΓI2 : v = (0, 0, 2), ΓO : outflow, ΓL : v = (0, 0, 0).

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

Incompressible test

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

The importance of density

The importance of density Computational domain:  Ω = (x , y , z) ∈ R3 | (x 2 + y 2 ) ≤ 0.52 , z ∈ [0, 5] . Species: O2 CO2 Boundary conditions: ΓI1 : v = (0, 0, 2), YO2 = 0.75, YCO2 = 0.25, θ = 1000, ΓI2 : v = (0, 0, 2), YO2 = 0.25, YCO2 = 0.75, θ = 1000, ΓO : outflow, ΓL : v = (0, 0, 2), YO2 = 0.75, YCO2 = 0.25, θ = 1000. A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

The importance of density

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

Second compressible test

Second compressible test Computational domain:  Ω = (x , y , z) ∈ R3 | (x 2 + y 2 ) ≤ 0.52 , z ∈ [0, 5] . Species: O2 CO2 Boundary conditions:   2 −y 2 ΓI1 : v = 0, 0, −x0.25 + 1 , YO2 = 0.75, YCO2 = 0.25, θ = 1000,   2 −y 2 ΓI2 : v = 0, 0, −x0.25 + 1 , YO2 = 0.25, YCO2 = 0.75, θ = 1200, ΓO : outflow,   2 −y 2 ΓL : v = 0, 0, −x0.25 + 1 , YO2 = 0.75, YCO2 = 0.25, θ = 1000. A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

Second compressible test

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

Second compressible test

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Numerical results

Second compressible test

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Ongoing research and conclusions

Ongoing research and conclusions

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Ongoing research and conclusions

Ongoing research

Divergence terms for low Mach number flows The equations of species transport and energy conservation in low Mach number flows include two new terms with respect to the incompressible equations: − wh div v

− wY div v

⇓

⇓ Z

Z −

wY div v dV X Z v ·e η dS ≈ − wY |Ni Ci

Nj ∈Ki

−

wh div v dV X Z ≈ − wh|Ni v ·e η dS

Γij

Ci

Nj ∈Ki

Γij

Source term treatment3,4 ? 3 A. Bermúdez, A. Dervieux, J.A. Desideri and M.E. Vázquez-Cendón "Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes". Comput.Methods in Appl.Mech.Eng. 155 49–72, 1998. 4 M.E. Vázquez-Cendón "Estudio de esquemas descentrados para su aplicación a las leyes de conservación hiperbólicas con términos fuente". PhD thesis, 1994. A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Ongoing research and conclusions

Conclusions

Conclusions Equations dependency Species transport (Y)

Navier-Stokes equations (π, w)

Turbulence model (k, ε)

Temperature (θ)

State equation

Energy conservation (h)

Models Incompressible Newtonian flow Low Mach number flow Methodology Hybrid projection method FV/FE

FV schemes 1st order CVC Kolgan-type Local ADER (ENO based)

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Acknowledgements

Acknowledgements This research was partially supported by Spanish MICINN project MTM2013-43745-R; by the Spanish MECD under grant FPU13/00279; by the Xunta de Galicia Consellería de Cultura Educación e Ordenación Universitaria under grant Axudas de apoio á etapa predoutoral do Plan I2C ; by Xunta de Galicia and FEDER under research project GRC2013014 and by Fundación Barrié under grant Becas de posgrado en el extranjero.

A FV/FE projection method for compressible low-Mach number flows

SHARK-FV, 2017

Thank you! S. Busto ([email protected]) www.usc.es/ingmat