computing of compressible flows using particle … flow... · compressible flow: physical aspects...

Computing of Compressible Flows using Particle Velocity Upwind Schemes

Prof. Nadeem Hasan

Dept. of Mechanical Engg.

Aligarh Muslim University

Contents

• Part I: Fundamental Aspects • Compressible flow: Physical aspects and governing equations

• Challenges in computing

• Existing approaches (generic features)

• Part II: PVU-M+ Scheme and its performance • Scheme

• Test Cases (Euler and Navier-Stokes)

• Part III: Conclusions and open issues

Part I: Fundamental Aspects

Compressible flow: Physical Aspects

• Significant Variations in density of a fluid particle caused by pressure variations - (eg. High speed flow) - Conventional Definition

In general , density=func(press., Temp.)

• Significant Variations in density of a fluid particle caused either by pressure or temperature variations – Extended Definition (Panton, Karamcheti …) (eq. low-speed flow with large-scale Temp. variations)

• Applications

Aerodynamics, Propulsion system components, Thermal Convection (large-scale Temp. variations)

Challenges in Computing Diverse physics

• Complex interaction between flow and compressibility waves discontinuous features like shocks, contacts and slip-lines.

• Interaction of shocks with domain boundaries (reflections) and with flow features like vortices and boundary layers.

• Complex energy dynamics : exchanges between internal and kinetic energy through the action of compression work, high dissipation

Numerical Stiffness

• Conflicting requirements for resolution of smooth flow features (vortices, boundary layers …) and discontinuous flow features (shocks, contacts, slip lines ….)

Challenges in Computing (Contd…)

• Scheme must be solution sensitive or solution adaptive • differentiate discontinuous features from smooth features

• adapt via changes in discretization strategy in order to resolve discontinous as well as smooth features without large build up of errors

• Numerical diffusion must be adaptively controlled so as to retain numerical stability near dicontinuous flow features without contaminating the physical diffusion in smooth flow zones

• Scheme should be relatively simple yet accurate and robust to capture the flow physics over a wide range of Mach numbers.

Governing Equations • Conservative forms of Continuity, Momentum and Energy equations

combined into a system or vector form

c nc c nc( ) ( ),

t x y

U F F G GJ

2

o o2

c nc c

2

o

T T2

oF

o o o

0

p 2 u( V) / 3u v

M Re xuvu

, , ,v uuv v

Re x yuh vh

( 1)MTk D

Re Pr x Re

F F G

o

nc

2

o o

2

oG

o o o

0

v u

Re x yu

, ,p 2 v( V) / 3 v

M Re yE

( 1)MTk D

Re Pr y Re

G U

2

o

2

o

2

o

0

0

Fr

( 1) vM

Fr

J

G

2 u 2 v v uD v u

3 x y x y

F

2 v 2 u v uD u v

3 y x x y

2

o TE e ( 1)M (V V) / 2, h E ( 1)p /

3 2

o oT p , e T, T (1 S / T ) (T S / T ), k

Perfect gas, const sp. Heats and Prandtl number

Existing Methods (Time-Marching)

• Coupled space-time approach (Less popular) Total truncation error in space and time is controlled

Order of discretization is increased by employing the governing equation

Eg. Lax-Wendroff family of schemes

• Decoupled space-time approach (More popular) first order Euler / a two-step predictor-corrector / multi-step RK method

spatial discretization:

(i) flux-based approach) : direct forward/backward/central discretization

MacCormack’s method

(ii) Wave-based approach (Type I) : Flux-Vector Splitting and discretization

Van Leer’s method, AUSM Family,....

(iii) Wave-based approach (Type II): Flux estimation through Riemann solver

Godunov’s method, HLL scheme, Roe scheme….

Wave-Based Methods: General Approach

• Total Flux (F) = Inviscid Flux (FI )+Viscous Flux (FV )

• Inviscid Flux Wave-based spatial discretization

• Viscous Flux Central discretization

Basis: Unsteady Euler equations are hyperbolic, represent wave dynamics

• Attempt to exploit wave dynamics for estimating Inviscid Fluxes

L R

i i+1

I I I I

int cell 1 L L R Ri

2

F F W (M )F W (M )F

W (M) 1.0, W (M) 0.0 M 1.0

W (M) 0.0, W (M) 1.0 M 1.0

W (M) W (M) 1.0 1 M 1

Splitting weight functions are not unique

I I

int cell 1 int celli

2

int cell L R

F F F(U )

U So ln of Riemann prob.(U , U )

Excellent resolution of discontinuities

Part II: PVU scheme and Test Cases

Particle Velocity Upwind Schemes: Origin

• Wave-based methods tend to be quite complex

• Little effort has been made towards development of flux-based methods

• Exploring the possibility of developing a simple flux-based method comparable in accuracy / robustness with state-of-the-art wave-based methods

• Towards development of a single unified method for handling low (nearly incompressible) as well as high Mach number flows

PVU scheme: Key ideas

• Total Flux (F) = Convective Flux (Fc )+Non-Convective(Fnc) • Components of Fc = (velocity)(transport property)

Note: This is different from the usual splitting into Inviscid and Viscous fluxes

• Time integration: Two-step Predictor-Corrector (similar to MacCormack’s method)

• Flux gradients: Fc Conservative central

Fnc Forward / Backward (as in MacCormack’s method)

• Intercell convective flux = weighted combination (first order upwind, third order upwind-biased, fourth order central interpolations)

• Weights are solution sensitive – identify smooth and discontinuous features, impart adaptability to discretization

The PVU-M+ scheme: Discretization details • A typical uniform 1-D mesh and a computational molecule surrounding the ith

node or grid point.

Computational molecule around ith node

For unsteady Euler equations in 1-D,

Predictor:

Corrector:

i 1/2 i 1/2 i 1 i

i 1/2 i 1/2 i 1 i

c n c n nc n nc n

* n tx x x x

F U F U F U F UU = U -

i 1/2 i 1/2 i i 1

i 1/2 i 1/2 i i 1

* n c * c * nc * nc *

n 1 t

2 2 x x x x

U U F U F U F U F UU

• The inter-cell numerical convective flux is expressed as

: Intercell velocity, : Intercell convective transport vector

Convective Transport Vector, Q :

, =interpolation[values at (i-1), i, (i+1), (i+2) nodes]

• Solution Sensitive Weight functions

A. Strong / weak convection: where

Wf 0, weak convection

Wf 1, strong convection

B. Smooth / discontinous flow feature:

c uF Q

The PVU-M+ scheme: Discretization details

i 1 2u i 1 2Qi 1/2 i 1/2 i 1 2

c u F Q

i 1 2[u i 1 2]Q

snf

sn

uW ,

u

i i 1sn

S S

u uu max , ,

U U

i 1 i i i 1

i i 1 i i i 1

i 1 i i i 1

var var var varZ if var var var var 0

var var var var

0 otherwise

i i 1(var) Max(Z , Z )

0 smooth feature

1 discontinuous feature

• Use of fully upwind schemes in a weak convection zone can generate oscillatory flow fields.

• smooth transition between higher order estimates in smooth solution regions and lower order estimates in neighborhood of discontinuities.

Note: A second estimate of intercell velocity is also obtained to achieve high order near discontinuities


f

f

w (C) (U) (C)

f

w (C) (U) (C)

f

u u W (u u )

W ( )Q Q Q Q

f

f f

W(1) (C) (C)

i 1 2

w w(L)

i 1 2

u u (u)(u u ),

( )( ).

Q Q U Q Q

f f(W ) (W )(2) (C)

i 1 2u u (u)(u u ).

• The fourth order central interpolation), third order upwind biased and first order upwind biased interpolation estimates of any discretely sampled function ‘f’ at midway location i+1/2 on a uniform grid

where, =(ui+ui+1)/2


C i 1 i i 1 i 2f 9f 9f ff ,

16

U i 1 i i 1 i 2 i 1 i i 1 i 2f 9f 9f f f 3f 3f ff sgn u ,

16 16

(L)

i i 1 i i 1f f f / 2.0 sgn u f - f / 2.0

u

• Higher order interpolations spurious oscillatory solution, especially near discontinuities

• To minimize the tendency, range boundedness criterion is employed

• The intercell estimates that satisfy the range boundedness criteria are chosen as the final estimates

• In the event the higher order interpolation estimates fail the range boundedness test, then intercell estimates = linear interpolation (vari, vari+1)

• If both estimates of intercell velocity satisfy the range boundedness criteria, then


i i 1 i 1/2 i i 1min , max ,Q Q Q Q Q i i 1 i 1/2 i i 1min u ,u u max u ,u

(1) (2)

i 1 2 i 1 2 i 1 2u (u u ) 2

A. Multi-dimensional scenario: weighted interpolated estimates for and resolve smooth as well as discontinuity like features quite well

• Numerical diffusion reduces in a 1D scenario , it is necessary to avoid interpolation across a shock in order to avoid numerical instability

B. One-Dimensional scenario :

-Threshold values of the function ψ are set to identify discontinuity like features.

Threshold values [0.7-0.9] are found to be suitable.

-presence of shock is confirmed through wave speeds. e.g for the 1-D Euler equations,

-Intercell estimates from one side of the cell interface give excellent results in 1D in shock regions ,

C. Non-Convective Fluxes : Viscous / thermal transport, dissipation terms – Central discretization

i i 1 i i 1

u c u c or u c u c .

i 1 2u i 1 2Q

i 1 2 iu u i 1 2 iQ Q


PVU-M+ scheme: Performance Test Cases

• 1D scalar hyperbolic problems Linear Advection Equation (LAE), Inviscid Burgers Equation (IBE)

• 1D Euler Equations Riemann Problems

• 2D Euler Equations Supersonic flow past a Forward Facing Step in a channel

Flow past a cylinder : Bifurcations with increase in free-stream Mach number

• 2D Navier-Stokes Re = 100, low Mach number thermal convection past square & circular

cylinders

Compressible laminar boundary layer over a flat plate at Re = 105

1D scalar hyperbolic probems

• Generic equation:

LAE: f(u)=cu, where c = const, models simple waves and contacts

- exact solutions are waveform preserving

- test cases reveal the effects of numerical dissipation and dispersion very clearly

IBE: f(u)=u2/2, models acoustic waves and shock formation

u f (u)0

t x

u(x, 0)= 1, for <1/3

= 0, for 1/3 1, t = 40, t=0.00015, N=320

LAE Test Cases

c=+1, Periodic domain [-1, +1]

u(x, 0)= -Sin(πx), t= 30, t=0.001

x

x

x

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1 exact

PVU-M+

PVU

u

x

-1 -0.5 0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2exact

PVU-M+

PVU

u

IBE Test Case : periodic domain [-1,+1]

u(x, 0)= 1, for <1/3

= -1, for 1/3 1

t = 0.3, t=0.0004, N=160

x

x

x

-1 -0.5 0 0.5 1

-1.5

-1

-0.5

0

0.5

1

1.5 exact

PVU-M+

PVU

u

1D Euler Test Cases: Riemann Problems

• Riemann Problems : Evolution of an initial imposed discontinuity and the subsequent gas flow under inviscid conditions Interaction of expansion waves, contact and shock discontinuities

• Characterized by an infinite domain, initially with a discontinuity at some point xo separating two portions of gas at different uniform conditions

(L, uL, pL ) (R, uR, pR )

x = x0

+ -

Riemann Problem: Test case

• Domain [0, +1], x0 = 0.5, t = 0.012

N = 200, time step = 10-5

• Initial data: (L, uL, pL)=(1.0, 0.0, 1000) x < 0.5 (R, uR, pR)=(1.0, 0.0, 0.001) x > 0.5

• End conditions: transmissive characteristic BC

Riemann Problem

Position

De

nsity

0 0.25 0.5 0.75 1

1

2

3

4

5

6Exact

PVU-M+

Van Leer

Position

Ve

locity

0 0.25 0.5 0.75 1

0

4

8

12

16

20

Exact

PVU-M+

Van Leer

Position

Pre

ssu

re

0 0.25 0.5 0.75 1

0

200

400

600

800

1000Exact

PVU-M+

Van Leer

density velocity pressure

Riemann Problem: density solutions

Position

De

nsity

0 0.25 0.5 0.75 1

1

2

3

4

5

6Exact

PVU-M+

HLL

PositionD

en

sity

0 0.25 0.5 0.75 1

1

2

3

4

5

6Exact

PVU-M+

AUSMPW+

Convergence characteristics

• rms error norm :

• E=(const)Nm

1/2L

2N

EX

0

1E u x, t u x, t dx

L

Variables

PVU-M+ Van Leer HLL AUSMPW+

m m m m

ρ -0.389 -0.362 -0.438 -0.336

u -0.496 -0.491 -0.525 -0.534

e -0.373 -0.334 -0.402 -0.208

CPU time Vs Accuracy

Log10

(E)

Lo

g1

0(C

PU

tim

e/s

tep

)

-0.4 -0.3 -0.2 -0.1

1.5

2

2.5

3

3.5PVU-M+

AUSMPW+

Van Leer

HLL

Error norm is for density soln. PVU-M+ outperforms AUSMPW+ PVU-M+ also likely to fare better than Van Leer’s scheme in obtaining high accuracy HLL is slightly better than PVU-M+ for this test case

2D Euler Test cases: Forward Facing Step Problem

M=3.0

3.0

1.0

0.6

0.2

0.2

walls

Involves : - both concave and convex sharp corners - complex flow features such as Mach as well as multiple Regular reflections, slip lines

Forward facing step: density contours

X

Y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

X

Y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

X

Y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

X

Y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

(a)

(b)

=0

=0.1

Mesh : Uniform, 240x80 Time step: 0.001

Forward facing step: Density contours (High Resolution), flowfield near corners

(a)

(b)

X

Y

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Y

0 1 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Y

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Mesh: Uniform, 480x160

u

Y

0.55 0.6 0.65

0.6

0.65

0.7

0.75

0.8

(a) (b) X

Y

0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

2D Euler Test: Inviscid Flow past a circular cylinder-Instabilities with Mach number • A test case with a sequence of Instabilities as Mach number is increased in

the range [0.38, 0.98]

• Spatio-temporal changes in flow patterns

steady periodic quasi-periodic periodic

Symmetric flow (no shocks) Non-symmetric, surface shocks, vortex-shedding

Non-symmetric, surface shocks, vortex-shedding wake shocks, wake vortex-shedding

• Investigated in detail by N. Botta (JFM, 1995; 301: 225-50)-High order Finite Volume, Godunov scheme

Inviscid Flow Past Circular cylinder

• Body-fitted Coordinates are employed

• Stretching parameters (a, p)=(0.2, 1.2)

• Infinite domain is truncated by a concentric circle of R = 20 (dia of cyl)

• A 181x211 structured grid is generated (rmin =0.006 (dia of cyl) )

• Governing equations are transformed in body-fitted coordinates

• Boundary conditions: Far-boundary : Characteristic BC’s

cylinder surface: Normal vel = 0, other variables extrapolated from interior

p p

p p

( a) ( a)

a a

0.5 0.5x e cos(2 ), y e sin(2 ).

e e

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=300

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=300

(a)

(b)

Steady Periodic

M = 0.38 M = 0.5

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=300

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=300

(a)

(b)

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=300

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=300

(a)

(b)

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=300

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=150

X

Y

-1 0 1 2-2

-1

0

1

2t=300

(a)

(b)

Cyclic flow pattern at M = 0.5 (density maps)

X

Y

0 1 2

-1

-0.5

0

0.5

1

t=278

X

Y

0 1 2

-1

-0.5

0

0.5

1

t=279

X

Y

0 1 2

-1

-0.5

0

0.5

1

t=280

X

Y

0 1 2

-1

-0.5

0

0.5

1

t=281

X

Y

0 1 2

-1

-0.5

0

0.5

1

t=282

X

Y

0 1 2

-1

-0.5

0

0.5

1

t=283

Temporal patterns

(a) (b)

)

(c)

(d)

)

t

CD

200 225 250 275 3000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

t

CD

200 225 250 275 3000.5

1

1.5

2

t

CD

300 325 350 375 400

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

t

CD

225 250 275 300

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

t

CD

325 350 375 4000.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

t

CD

325 350 375 4000.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

(e) (f)

M = 0.6, periodic M = 0.7, Quasi-periodic M = 0.8, Quasi-periodic

M = 0.9, Quasi-periodic M = 0.95, Quasi-steady M = 0.98, Quasi-steady

Spatial Patterns

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=300

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=400

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=400

-1 0 1 2 3 4

-2

-1

0

1

2

t=400

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=300

X

Y

-1 0 1 2 3 4

-2

-1

0

1

2

t=300

M∞ = 0.6 M∞ = 0.7 M∞ = 0.8

M∞ = 0.9 M∞ = 0.95 M∞ = 0.98

Comparison with Numerical Data: Drag and Strouhal number data

freestream mach no.

CD

0.4 0.5 0.6 0.7 0.8 0.9 1

1.25

1.5

1.75

2

present

Botta

freestream mach no.

str

ou

ha

ln

o.

0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5present

Botta

(a) (b)

Inviscid Supersonic flow (circular cylinder):

X

Y

-1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

XY

-1 -0.5 0 0.5-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

M =3.0 M =10.0

Stagnation line data: (M = 3.0)

X

Cp

-2 -1.75 -1.5 -1.25 -1 -0.75 -0.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2AUSM

RoeFDS

PVU-M+

X-2 -1.75 -1.5 -1.25 -1 -0.75 -0.5

-0.2

0

0.2

0.4

0.6

0.8

1

1.2PVU-M+

AUSM

Roe-FDS

2D Navier-Stokes Test Cases: Thermal Convection • Thermal convection with large scale heating effects (low Mach, low Re)-Non-

Boussinesq approach

- at low Mach numbers ( < 0.5), absolute pressure scaling does not give good results

- The scaling usually employed in incompressible flow models gives much better results,

- For incorporating effects of gravity (buoyancy),

Po local pressure that would prevail under thermal& mechanical

equilibrium.

2

o o op (P P ) / U

Governing Equations: Non-Boussinesq model

2

22

F

u

2 uu p ( ) / 3

Re x

,v uuv

Re x y

T ( 1)MuE k ( 1)M up D

RePr x Re

V

F

2

22

G

v

v uuv

Re x y

,2 vv p ( V) / 3

Re y

T ( 1)MvE k ( 1)M vp D

Re Pr y Re

G

2

2

0

0

,(1- ) / Fr

M-1 (1- ) v - -1 .

Fr

J

V

For large-scale temperature variations: -Transport property variations -sp. Heat variations

Flow past bluff-bodies (large temp. diff.)

x

U T

,T∞

g

y

d

TW≠T

x

U T

g

y

d

T W T

Working fluid is a perfect gas:

22 2 2

2 3

v 1 2 3

2 3 431 2

γ(γ 1) 1 + γ M pE e M u v , ρ = ,

2 T

C 1 C (T 1) C (T 1) C (T 1) ,

CC Ce = T + T 1 + T 1 + T 1 ,

2 3 4- - -

3

2

2

1 + μ = T ,

T +

k = A + BT + CT .

Flow past bluff bodies

• Dimensionless parameters

-governing eqs. (, Re, M, Pr, Fr)

-Thermal BC at bluff body, specified Temp:

where : Over-heat ratio (case of heating)

- Sutherland law Constant:

- Geometric parameters:

• For small scale temp. variations, Richardson number can be defined as:

Ri = (for perfect gases)

WT 1

S / T

,

W W

2 2 2

g (T T )d (T T )gd

TU U Fr

W(T T )

T

• Numerical aspects: Body-fitted coordinates and structured grid

-Location of far-boundary: 60 (edge of sq. cylinder / dia of circ. cylinder)

Grid Structure

X

Y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) (b)

X

Y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Y

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) (b) Grid: 241 x 325 (min spacing = 0.003) Grid: 181 x 394 (min spacing = 0.003)

Mixed Convection from Sq. Cylinder • Parameters: = 1.4, Re = 100, M = 0.1, Pr = 0.7, Fr = 1.0, = 0, α = 0 and σ = 110 / 298

= 0.369, 0 ≤ ε ≤ 2 (steps of 0.2)

-2 0 2 4

-2

0

2

4

Fr = 1

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

4

Fr = 1.0

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

4

Fr = 1.0

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

4

Fr = 1.0

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

4

Fr = 1.0

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

4

Fr = 1.0

-3 -2 -1 0 1 2 3

-2

-1

0

1

2

3

4

Fr = 1.0Vortex-shedding suppression is captured by NB model and PVU-M+ scheme

Comparison with Experiments: Forced Convection from Circular Cylinder

• Parameters: = 1.4, M = 0.1, Pr = 0.7, α = 0 and σ = 110 / 298 = 0.369, Fr = (1/Fr2 terms are dropped), Re = [75, 100, 125, 150], ε = [0.1, 0.5]

Ref

50 75 100 125 1503.2

3.6

4

4.4

4.8

5.2

5.6

6

Present

Present

Collis and Williams (1959)


Nuf

(a)

+

+

+

+

+

+

+

++

+

x

x

x

x

x

x

x

x

xx

Re75 100 125 150

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0.19Present

Wang et al. (2000)

Present

Wang et al. (2000)

+

x

St

(b)

+

+

+

+

+

+

+

+

+

x

x

x

x

x

x

xx

x

x

Ref

50 60 70 80 90 100 110 120 130 140

3.6

4

4.4

4.8

5.2

5.6

6

6.4Present

Wang and Travnicek (2001)

Present


+

x

Nuf

Ref

50 75 100 125 1503.2

3.6

4

4.4

4.8

5.2

5.6

6

Present

Present



Nuf

(a)

+

+

+

+

+

+

+

++

+

x

x

x

x

x

x

x

x

xx

Re75 100 125 150

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0.19Present

Wang et al. (2000)

Present

Wang et al. (2000)

+

x

St

(b)

+

+

+

+

+

+

+

+

+

x

x

x

x

x

x

xx

x

x

Ref

50 60 70 80 90 100 110 120 130 140

3.6

4

4.4

4.8

5.2

5.6

6

6.4Present


Present


+

x

Nuf

w ff W f

f f

kNu Nu , Re Re

k

Compressible Boundary Layer over a flat plate: Re = 105

• Cartesian Coordinates, Domain square [0, 1]x[0, 1]

• Grid: Non-uniform, 226x335, xmin = ymin = 10-4

xmax = 0.035, ymax = 0.015

XY

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Thermal BC at plate: Isothermal wall, Tw= T

Far boundary

wall

U, T

Characteristic BC at inflow, outflow, far boundary

Parameters: M = 0.7, 2.0, 3.0 Pr = 0.7, =1.4 (air )

Relative pressure scaling resolves the BL more accurately

Compressible B.L: Results (profiles x = 0.6)

u-velocity

y

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

Mach = 0.7

Mach = 2

Mach = 3

v-velocityy

0 0.0025 0.005 0.00750

0.01

0.02

0.03

0.04

0.05Mach = 0.7

Mach = 2

Mach = 3

Compressible B.L : Results (profiles x = 0.6)

dimensionless temperature

y

1 1.1 1.2 1.30

0.01

0.02

0.03

0.04

0.05

Mach = 0.7

Mach = 2

Mach = 3

dimensionless densityy

0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

Mach = 0.7

Mach = 2.0

Mach = 3.0

Compressible B.L : Self-Similarity,

u-velocity0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0

1

2

3

4

5

6

7

8

9

10

x = 0.3

x = 0.6

x = 0.9

u-velocity0 0.25 0.5 0.75 1

0

1

2

3

4

5

6

7

8

9

10x = 0.3

x = 0.6

x = 0.9

M = 0.7 M = 3.0

1 2

yRe

x

Compressible B.L : Wall shear and Drag

Mach No CD (present) CD (van Driest)

0.7 0.00424 0.00413

2.0 0.00420 0.00410

3.0 0.00410 0.00405

x

Cf

0 0.25 0.5 0.75 110

-3

10-2

10-1

Mach = 0.7

Mach = 2.0

Mach = 3.0

Part III: Conclusions & Open questions

Conclusions • PVU-M+ scheme demonstrates good overall accuracy and robustness

• Capability of resolving wide ranging flow physics from low Mach (0.1) to High Mach number (=10) flows

• Excellent shock resolving capabilities

• Simplicity of implementation

- relies on simple interpolation / finite difference rules

- Avoids large number of flux-limiters and adjustable weight functions with parameters

• The present scheme can deliver comparable and even better performance than some of the established wave-based schemes

Open Questions / Issues

• Performance in computing of compressible turbulent flows (LES, DNS approaches)

• A finite volume version for unstructured grids needs to be developed

• Suitability of the method for acoustic computations

• Scalability of the algorithm under parallel computing environments

Publications:

1. N. Hasan, S.M. Khan, F. Shameem, “A New Flux-Based Scheme for Compressible Flows”, Comp. Fluids (in press), doi:10.1016/j.compfluid.2015.06.026

2. A. Qamar, N. Hasan, S. Sanghi, “New scheme for the computation of compressible flows”, AIAA J.; 44(5): 1025-1039 (2006).

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