Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Navier-Stokes Equations and Finite Volume Methods in Computational Fluid Dynamics, Slides of Fluid Dynamics

Biju Patnaik University of TechnologyFluid Dynamics

An outline for a lecture on the navier-stokes equations and finite volume methods in computational fluid dynamics (me 692). Topics covered include finite-volume convection algorithms, false diffusion, solving the navier-stokes equations, conservative and transportive schemes, and the staggered grid method. The document also includes a review of the integrated pde, example problem, and upwind differences.

Typology: Slides

2013/2014

Uploaded on 02/01/2014

savitri_122
savitri_122 🇮🇳

4.6

(14)

184 documents

1 / 8

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Navier-Stokes Introduction March 1, 2010
ME 692 – Computational Fluid Dynamics 1
Introduction to Solution of Navier
Introduction to Solution of Navier-
-
Stokes Equation
Stokes Equation
Larry Caretto
Mechanical Engineering 692
Computational Fluid Dynamics
March 1, 2010
2
Homework for March 3
• Download the Excel workbook from the
course web site for the sample
convection problem with Pecell = 1.25
– Shows results for Central and Upwind on
separate worksheets
• Add similar worksheets to get results for
Hybrid, Power Law, and QUICK
• Add error results for these algorithms to
the error chart
• Any questions?
3
Outline
• Review finite-volume convection
– Central, upwind, power law, QUICK, TVD
• False diffusion
• Solving the Navier-Stokes Equations
– Approaches
–Grids
– Pressure terms and the need for staggered
grids
– Derivation of momentum equations
4
Review Algorithm Properties
• Conservative schemes – conserve
properties in finite difference equations
– Requires exit flux from one face to be
same as input flux in adjacent cell
• Transportive schemes – have correct
balance between diffusion and
convection
• Accuracy – need schemes that have a
good truncation error
5
Review Algorithm Properties II
• Limit on coefficient magnitude for
iteration schemes (boundedness)
– Absolute value of diagonal coefficient must
be greater than the sum of absolute values
of all other coefficients
• For simple equations here |aP| ≥|aE| + |aW|
– Deferred correction separates coefficients
into two parts
• Adjus tment leaves |aP| ≥|aE| + |aW|
• Places part removed from adjusted coefficients
into source term
6
Review Convection Terms
• Steady equation with
convection and diffusion
terms in one dimension dx
d
dx
d
dx
ud
ϕ
Γ=
ϕ
ρ
0=
ρ
dx
ud
• Steady continuity
equation in one dimension
• Apply finite volume approach to
integrate small volume
dV
dx
d
dx
d
dV
dx
ud ∫∫
ϕ
Γ=
ϕ
ρ
0=
ρ
∫dV
dx
ud
pf3
pf4
pf5
pf8

Partial preview of the text

Download Navier-Stokes Equations and Finite Volume Methods in Computational Fluid Dynamics and more Slides Fluid Dynamics in PDF only on Docsity!

Introduction to Solution of NavierIntroduction to Solution of Navier--

Stokes EquationStokes Equation

Larry Caretto Mechanical Engineering 692 Computational Fluid Dynamics

March 1, 2010

2

Homework for March 3

  • Download the Excel workbook from the course web site for the sample convection problem with Pecell = 1. - Shows results for Central and Upwind on separate worksheets
  • Add similar worksheets to get results for Hybrid, Power Law, and QUICK
  • Add error results for these algorithms to the error chart
  • Any questions?

3

Outline

  • Review finite-volume convection
    • Central, upwind, power law, QUICK, TVD
  • False diffusion
  • Solving the Navier-Stokes Equations
    • Approaches
    • Grids
    • Pressure terms and the need for staggered grids
    • Derivation of momentum equations 4

Review Algorithm Properties

  • Conservative schemes – conserve properties in finite difference equations - Requires exit flux from one face to be same as input flux in adjacent cell
  • Transportive schemes – have correct balance between diffusion and convection
  • Accuracy – need schemes that have a good truncation error

5

Review Algorithm Properties II

  • Limit on coefficient magnitude for iteration schemes (boundedness) - Absolute value of diagonal coefficient must be greater than the sum of absolute values of all other coefficients - For simple equations here |aP | ≥ |aE | + |aW | - Deferred correction separates coefficients into two parts - Adjustment leaves |aP | ≥ |aE | + |aW | - Places part removed from adjusted coefficients into source term 6

Review Convection Terms

  • Steady equation with convection and diffusion terms in one dimension dx

d dx

d dx

d u ϕ = Γ ρϕ

= 0 ρ dx

  • Steady continuity du equation in one dimension
  • Apply finite volume approach to integrate small volume

dV dx

d dx

d dV dx

du ∫ ∫

ϕ = Γ ρ ϕ = 0 ρ ∫ (^) dxdV

du

7

Review Integrated PDE

  • Constant area result
    • Define F = ρu and D = Γ/δx

Fe ϕ e − Fw ϕ w = De ( ϕ E −ϕ P ) − Dw (ϕ P −ϕ W )

  • Different approaches for φe and φw
    • Central difference, upwind, hybrid, power- law, QUICK, TVD
    • All get relations among neighbor nodes
      • Three nodes for all but QUICK
      • Special treatment for boundary nodes

aW ϕ W − aP ϕ P + aE ϕ E = 0 8

Review Example Problem

  • Constant ρ, u, and Γ with φ = φ 0 at x = 0 and φ = φL at x = L

dx

d dx

d dx

ud dx

d dx

d dx

d u ϕ

ϕ Γ

ρ ⇒ ϕ = Γ ρϕ

1

1 0

0 −

= − ϕ −ϕ

ϕ −ϕ Γ

ρ

Γ

ρ

uL

ux

L e

x e

Pe = ρuL/Γ Pecell = ρuδx/Γ = F/D

  • Exact solution below with plot on next slide

9

Exact Solution

0

0.

0.

0.

0.

0.

0.

0.

0.

0.

1

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

( ξ

-^ ξ^0

)/( ξ L

-^ ξ^0 )^ Pe = - Pe = - Pe = - Pe = - Pe = 1 Pe = 2 Pe = 5 Pe = 10

10

Review Central Difference

  • Here δx, ρu and Γ are constants
    • F (^) e = F (^) w = ρu = F De = D (^) w = Γ / δx = D 0 2 2 2 ⎟ϕ = ⎠

⎜ ⎞ ⎝

⎟ϕ − ϕ +⎛^ − ⎠

⎜ ⎞ ⎝

ϕ − ϕ + ϕ =⎛^ + W W PP E E W P E

D F F D a a a F

  • Boundary conditions at x = 0 and x = L

( F D ) left

F

D D

F

⎟ϕ =− + ϕ ⎠

ϕ+⎛^ − ⎟ ⎠

− ⎛^ + 2

1 2

D F N D F ⎟ϕ N =− ( D − F )ϕ right

⎠

⎜ ⎞ ⎝

⎟ϕ −⎛^ − ⎠

⎜ ⎞ ⎝

⎛ (^) + 2 −^232 −^12

| -----●----- | -----●----- | -----●----- | -----●----- | -----●----- | -----●----- | left 1 2 3 4 5 6 right

11

Review Upwind Differences

  • Computational formulas aW = Dw +max( Fw , 0 ) aE = De +max(− Fe , 0 ) aP = aE + aW + Fe − F w

| ----------●---------- | ----------●---------- | ----------●---------- | W w P e E

  • Left boundary

− ( a E + aW *+ Fe − Fw )ϕ P + aE ϕ E =− aW *ϕ left

aW *^ = 2 Dw +max( Fw , 0 )

  • Right boundary

aW ϕ W − ( a E *+ aW ++ Fe − Fw )ϕ P =− aE *ϕ right

a * E^ = 2 De +max(− Fe , 0 )

12

Review Hybrid Difference

  • Computational Formulas

⎥ ⎦

⎤ ⎢ ⎣

⎡ ⎟ ⎠

⎜ ⎞ ⎝

= − ⎛^ + ⎥ ⎦

⎤ ⎢ ⎣

⎡ ⎟ ⎠

⎜ ⎞ ⎝

= ⎛^ + , 0 2 , 0 max , 2 W max^ w , w w E e e e

F a F D F a F D aP = aE + aW + Fe − Fw

  • Left boundary

− ( a E + aW *+ Fe − Fw )ϕ P + aE ϕ E =− aW *ϕ left

  • Right boundary

aW ϕ W − ( a E *+ aW + Fe − Fw )ϕ P =− aE *ϕ right

aW * = max[ 2 Dw , ( 2 Dw + Fw )]

aE * = max[ 2 De , ( 2 De − Fe )]

TVD Flux Limiters

0

1

2

3

0 1 2 3 4 5 6 7 8 r

ψ (r)

Van Leer Van Albada min-mod SUPERBEE UMIST^20

False Diffusion

  • Upwind differencing causes errors similar to having a “diffusion” coefficient that is too large
  • Causes smearing of results
  • Especially noticeable in flows with sharp gradients and shock waves
  • Effect is reduced if flow is aligned with grid (not always possible to do this)
  • Different from artifical diffusion

21

Navier-Stokes Equations

  • Continuity and x-momentum

= 0 ∂

∂

∂

∂

∂

∂

∂

∂ z

w y

v x

u t

ρ ρ ρ ρ

S ( u^ ) z

u y z

u x y

u x x

P

z

wu y

vu x

uu t

u

⎥⎦

⎤ ⎢⎣

⎡ (^) − Δ ∂

∂

∂

∂ ∂

∂

∂

∂ ∂

∂

∂

∂ ∂

∂ = + ) 3

2 () ρ μ μ μ (κ μ x x

w x z

v x y

u x S u Bx

22

y-momentum Equation

S ( v^ ) z

v y z

v x y

v y x

P

z

wv y

vv x

uv t

v

()

μ κ μ

ρ μ μ

y y

w z

y

v y y

u x

S v By

23

z-momentum Equation

S (^ w^ ) z

w y z

w x y

w z x

P

z

ww y

vw x

uw t

w

()

μ κ μ

ρ μ μ

z z

w z

z

v z y

u x

S w Bz

24

Navier-Stokes Equations

  • Continuity and momentum equations
  • Up to now we have been assuming that the velocity field was known and we could find the general variable, φ
  • This background is necessary for solving Navier-Stokes, but now we have to solve for φ = u, v, and w
  • This gives a set of nonlinear equations (e.g., uφ becomes uu for x-momentum)

25

Navier-Stokes Equations II

  • Have to find way to solve nonlinear equations
  • Basic approach requires “outer” iteration process - Assume values for u, v, and w - Use these values to compute the convec- tion/diffusion coefficients a (^) E, aN , etc. - Solve finite difference forms of the Navier- Stokes equations for new values of u, v, and w using these “old” aE, aN , etc. 26

Navier-Stokes Equations III

  • Once new values of u, v, and w are known update aE, aN , etc. iterate again
  • Consider steady-state flows now and transient flows later
  • Some transient methods can use nonlinear terms at old time step to get new values for aE, aN, etc.
  • For steady flows the iterations on the nonlinear terms becomes part of the overall iteration process

27

Finding Pressure and Density

  • For compressible flows we solve continuity and momentum for density, u, v, and w - Get pressure from equation of state (e.g., p = ρRT) for compressible flows
  • For incompressible flows find u, v, w, and p to satisfy three momentum equations and continuity - Density is an input parameter or depends on variables other than local pressure 28

Incompressible Flows

  • Mach number is low (< ~0.3)
  • Density (e.g., ρ = p/RT) may depend on mean, but not local pressure
  • Can have equations like ρ = ρ 0 ( 1 + βT ) for where β = –(1/ρ)(∂ρ/∂T) (^) P
  • Density may be constant, but need not be for incompressible flows - Furnace flows a good example of this
  • Basic idea is that density does not depend on local pressure

29

Navier-Stokes Problems

  • Compressible flows
    • Solve continuity and momentum for three velocity components and density
    • Get pressure from equation of state
  • Incompressible flows
    • Mach number is low
    • Density is a problem input, often related to temperature (may or may not be constant)
    • Solve continuity and momentum for three velocity components and pressure 30

The Steady 2D Problem

= 0 ∂

∂ρ

∂

∂ρ y

v x

u

S ( u^ ) y

u x y

u x x

P

y

vu x

uu

∂

μ ∂

μ ∂

∂ρ

∂

∂ρ

S ( v^ ) y

v x y

v y x

P

y

vv x

uv

∂

μ ∂

μ ∂

∂ρ

∂

∂ρ

  • Continuity and momentun equations
    • Have x and y direction convection-diffusion
    • Now have source term and pressure gradient

37

Finite Volume Equations II

  • Integration of pressure terms

( p p ) A ( p p )( y y ) z

x x A x x

p p dV x

p

IJ I J iJ IJ I J j j

I I iJ I I

IJ I J V iJ = − = − − Δ

− −

− ⎟⎟ ≈ ⎠

⎞ ⎜

⎜ ⎝

⎛ ∂

∂

− − +

− −

− Δ

∫

1 1 1

1 1

1

( p p ) A ( p p )( x x ) z

y y A y y

p p dV y

p

IJ IJ Ij IJ I J i i

J J Ij J J

IJ IJ V Ij = − = − − Δ

− −

− ⎟⎟ ≈ ⎠

⎞ ⎜

⎜ ⎝

⎛ ∂

∂

− − +

− −

− Δ

∫

1 1 1

1 1

1

38

Finite Volume Equations III

  • Have similar equations for u and v
  • b represents integrated source term
  • Note that aK coefficients vary from node to node and are different for u and v

( 1 ) ()

1 1 1 1 u iJ i J iJ iJ

N iJ S iJ E i J W i J P iJ

p p A b

a u au a u a u a u

−

  • − + −

( 1 ) ()

1 1 1 1 v Ij Ij Ij Ij

N Ij SiJ E I j W I j P Ij

p p A b

a v a v a v a v a v

−

  • − + −

1

( ) iJ +

u

a N iJu

1

( ) Ij +

v

aN Ijv

39

Control Volume for u (e face) PIJ+

PI-1J

vIj

uiJ

PIJ-

PIJ ui+1J

vIj+

vI-1j

vI-1j+

uiJ-

ui-1J

equations for e face at IJ

⎥ ⎦

⎤ ⎟ ⎠

⎜ ⎞ ⎝

+⎛^ ρ +ρ

⎢ ⎣

⎡ ⎟ ⎠

⎞ ⎜ ⎝

⎛ ρ +ρ

= + =

−

iJ I J IJ

IJ I J i J

e l J iJ

u

u

F F F

2

2 2

1

2

1

(^11)

1

i i

e De (^) x − x

  • 1

uiJ+

40

Control Volume for u (w face) PIJ+

PI-1J

vIj

uiJ

PIJ-

PIJ ui+1J

vIj+

vI-1j

vI-1j+

uiJ-

ui-1J

equations for w face at I-1J

⎥ ⎦

⎤ ⎟ ⎠

⎜ ⎞ ⎝

+⎛^ ρ +ρ

⎢ ⎣

⎡ ⎟ ⎠

⎞ ⎜ ⎝

⎛ ρ +ρ

= + =

−

− − −

−

iJ I J IJ

I J I J i J

w l J iJ

u

u

F F F

2

2 2

1

2

1

(^211)

1

1

1 −

− −

i i

I J Dw (^) x x

uiJ+

41

Control Volume for u (n face) PIJ+

PI-1J

vIj

uiJ

PIJ-

PIJ ui+1J

vIj+

vI-1j

vI-1j+

uiJ-

ui-1J

equations for n face at yj+

⎥ ⎦

⎤ ⎟ ⎠

⎞ ⎜ ⎝

⎛ ρ +ρ

⎢⎣

⎡ ⎟ + ⎠

⎜ ⎞ ⎝

⎛ ρ +ρ

=

=

− − + − +

  • −+

(^11111)

1 1

1 1 1

2

2 2

1

2

I J I J I j

Ij IJ IJ

Ij I j n

v

v

F F F

( (^) J J )

I J I J IJ IJ Dn (^) y − y

− − + + 1

1 1 1 1 4

uiJ+

42

Control Volume for u (s face) PIJ+

PI-1J

vIj

uiJ

PIJ-

PIJ ui+1J

vIj+

vI-1j

vI-1j+

uiJ-

ui-1J

equations for n face at yj+

⎥ ⎦

⎤ ⎟ ⎠

⎞ ⎜ ⎝

⎛ ρ +ρ

⎢⎣

⎡ ⎟ + ⎠

⎜ ⎞ ⎝

⎛ ρ +ρ

=

=

− − − −

−

−

(^1111)

1

1

2

2 2

1

2

I J I J I

Ij IJ IJ

Ij I j s

v

v

F F F

( 1 )

1 1 1 1 (^4) −

− − − − −

J J

I J I J IJ IJ Ds (^) y y

uiJ+

43

Where to Next

  • Have similar equations to give various F and D terms for v control volume
  • Get finite volume representation of continuity by integration over control volume centered about pIJ s
  • Substitute finite difference momentum equations into finite difference continuity equation to get finite difference equation for pressure
  • Develop solution procedure for u, v, p