non dimensionalization

Non-dimensionalisation of the Navier-Stokes equations

Michal Kopera

Centre for Scientific Computing

24 January 2008

Michal Kopera Non-dimensionalisation of the Navier-Stokes equations

Outline

Previously on Research Seminar ...

Navier-Stokes equations (vector and tensor notation)

Different forms of Navier-Stokes equations

Divergent

Dissipative

Skew-symmetric

Rotational

Non-dimensionalization

Dimensionless variables

Derivations of dimensionless equations

Choosing the reference values

Previously on Research Seminar...

Continuity equation

∂t+∂(ρu)

∂x+∂(ρv)

∂y+∂(ρw)

∂z= 0

Assumptions

Infinitesimal control volume, but big enough to treat the fluid as acontinuum

ρ = const.

Continuity equation

∂t+∂(ρu)

∂x+∂(ρv)

∂y+∂(ρw)

∂z= 0

Assumptions

ρ = const.

Continuity equation

∂t+∂(ρu)

∂x+∂(ρv)

∂y+∂(ρw)

∂z= 0

Assumptions

ρ = const.

Continuity equation for incompressible fluid

∂x+∂v

∂y+∂w

∂z= 0

∇~V = 0

∂xi= 0

∂x+∂v

∂y+∂w

∂z= 0

∇~V = 0

∂xi= 0

∂x+∂v

∂y+∂w

∂z= 0

∇~V = 0

∂xi= 0

Linear momentum equation

ρ∂~V

∂t+ ρ(~V · ∇)~V = ρ~g −∇p +∇τij

ρ∂ui

∂t+ ρuj

∂xj= ρgi −

∂xi+∂τij∂xj

Assumptions

Newtonian fluid: τij = µ(∂ui∂xj

+∂uj

ρ∂~V

∂t+ ρ(~V · ∇)~V = ρ~g −∇p +∇τij

ρ∂ui

∂t+ ρuj

∂xj= ρgi −

∂xi+∂τij∂xj

Assumptions

+∂uj

ρ∂~V

∂t+ ρ(~V · ∇)~V = ρ~g −∇p +∇τij

ρ∂ui

∂t+ ρuj

∂xj= ρgi −

∂xi+∂τij∂xj

Assumptions

+∂uj

Linear momentum equation for Newtonian fluid

∂t+ (~V · ∇)~V = ~g − 1

ρ∇p + ν4~V

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

Linear momentum equation for Newtonian fluid

∂t+ (~V · ∇)~V = ~g − 1

ρ∇p + ν4~V

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

Energy equation

dt+ p(∇ · ~V ) = ∇(k · ∇T ) + Φ

Assumptions

Fourier law: ~q = −k∇T

du ≈ cvdT ≈ cpdT

cv , cp, µ, k , ρ ≈ const.

Φ = 0

Energy equation

dt+ p(∇ · ~V ) = ∇(k · ∇T ) + Φ

Assumptions

Φ = 0

Energy equation

dt+ p(∇ · ~V ) = ∇(k · ∇T ) + Φ

Assumptions

Φ = 0

Energy equation

dt+ p(∇ · ~V ) = ∇(k · ∇T ) + Φ

Assumptions

Φ = 0

Energy equation for an incompressible flow without viscous dissipation

∂t+ (~V · ∇)T =

ρcp4T

∂t+ uj

∂xj=

∂xj∂xj

Energy equation for an incompressible flow without viscous dissipation

∂t+ (~V · ∇)T =

ρcp4T

∂t+ uj

∂xj=

∂xj∂xj

Different forms of the Navier-Stokes equation

convective form

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

divergent form

∂t+∂(uiuj)

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

skew-symmetric form

2uj∂ui

∂xj+

∂(uiuj)

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

convective form

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

divergent form

∂t+∂(uiuj)

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

skew-symmetric form

2uj∂ui

∂xj+

∂(uiuj)

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

convective form

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

divergent form

∂t+∂(uiuj)

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

skew-symmetric form

2uj∂ui

∂xj+

∂(uiuj)

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

rotational form

∂t+ (∇× ~V )× ~V +∇

2|~V |2

)= ~g − 1

ρ∇p + ν4~V

~ω = ∇× ~V

P = p +1

2ρ|~V |2

∂t+ ~ω × ~V = ~g − 1

ρ∇P + ν4~V

rotational form

∂t+ (∇× ~V )× ~V +∇

2|~V |2

)= ~g − 1

ρ∇p + ν4~V

~ω = ∇× ~V

P = p +1

2ρ|~V |2

∂t+ ~ω × ~V = ~g − 1

ρ∇P + ν4~V

rotational form

∂t+ (∇× ~V )× ~V +∇

2|~V |2

)= ~g − 1

ρ∇p + ν4~V

~ω = ∇× ~V

P = p +1

2ρ|~V |2

∂t+ ~ω × ~V = ~g − 1

ρ∇P + ν4~V

convective term in tensor notation

(∇× ~V )× ~V +∇(

2|~V |2

)→ uj

(∂ui

∂xj−∂uj

)rotational form in tensor notation

∂t+ uj

(∂ui

∂xj−∂uj

)= gi −

∂xi+ ν

∂2ui

∂xj∂xj

convective term in tensor notation

(∇× ~V )× ~V +∇(

2|~V |2

)→ uj

(∂ui

∂xj−∂uj

)rotational form in tensor notation

∂t+ uj

(∂ui

∂xj−∂uj

)= gi −

∂xi+ ν

∂2ui

∂xj∂xj

Summary of the Navier-Stokes equation formulations

convective ∂ui∂t + uj

∂ui∂xj

= gi − 1ρ∂p∂xi

+ ν ∂2ui∂xj∂xj

divergent ∂ui∂t +

∂(uiuj )∂xj

skew-symmetric ∂ui∂t + 1

2uj∂ui∂xj

+ 12∂(uiuj )∂xj

rotational ∂ui∂t + uj

(∂ui∂xj− ∂uj

)= gi − 1

ρ∂P∂xi

Non-dimensionalization - Dimensionless variables

a - variable which has a dimension (unit) (length, velocity)

A - reference constant (eg. length of a football pitch, free streamvelocity)

a∗ = aA - dimensionless variable

Non-dimensionalization - Continuity equation

Continuity equation

∂xi= 0

u∗i = uiU x∗i = xi

Non-dimensional continuity equation

∂u∗i∂x∗i

Continuity equation

∂xi= 0

∂u∗i∂x∗i

Continuity equation

∂xi= 0

∂u∗i∂x∗i

Non-dimensionalization - Momentum equation

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

L t∗ = t UL p∗ = p

Non-dimensional momentum equation

∂u∗i∂t∗

+ u∗j∂u∗i∂x∗j

U2− ∂p∗

∂x∗i+

∂2u∗i∂x∗j ∂x∗j

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

L t∗ = t UL p∗ = p

∂u∗i∂t∗

U2− ∂p∗

∂x∗i+

∂t+ uj

∂xj= gi −

∂xi+ ν

∂2ui

∂xj∂xj

L t∗ = t UL p∗ = p

∂u∗i∂t∗

U2− ∂p∗

∂x∗i+

∂u∗i∂t∗

U2− ∂p∗

∂x∗i+

Dimensionless parameters

Reynolds number: Re = ULν (always important)

Froude number: Fr = U2

gL (only if there is a free surface)

Non-dimensional momentum equation with dimensionless parameters

∂u∗i∂t∗

Fr− ∂p∗

∂x∗i+

Non-dimensionalization - Energy equation

Energy equation

∂t+ uj

∂xj=

∂xj∂xj

L t∗ = t · t0 T ∗ = T−T0T1−T0

Non-dimensional energy equation

T1 − T0

∂T ∗

∂t∗+

L(T1 − T0)u∗j

∂T ∗

∂x∗j=

T1 − T0

∂T ∗

∂x∗j ∂x∗j

Energy equation

∂t+ uj

∂xj=

∂xj∂xj

L t∗ = t · t0 T ∗ = T−T0T1−T0

T1 − T0

∂T ∗

∂t∗+

L(T1 − T0)u∗j

∂T ∗

∂x∗j=

T1 − T0

∂T ∗

∂x∗j ∂x∗j

Energy equation

∂t+ uj

∂xj=

∂xj∂xj

L t∗ = t · t0 T ∗ = T−T0T1−T0

T1 − T0

∂T ∗

∂t∗+

L(T1 − T0)u∗j

∂T ∗

∂x∗j=

T1 − T0

∂T ∗

∂x∗j ∂x∗j

T1 − T0

∂T ∗

∂t∗+

L(T1 − T0)u∗j

∂T ∗

∂x∗j=

T1 − T0

∂T ∗

∂x∗j ∂x∗j

Reynolds number: Re = ULν

Prandtl number: Pr =µcp

Strouhal number: St = LUt0

Non-dimensional energy equation with dimensionless parameters

St∂T ∗

∂t∗+ u∗j

∂T ∗

∂x∗j=

∂T ∗

∂x∗j ∂x∗j

T1 − T0

∂T ∗

∂t∗+

L(T1 − T0)u∗j

∂T ∗

∂x∗j=

T1 − T0

∂T ∗

∂x∗j ∂x∗j

Reynolds number: Re = ULν

Prandtl number: Pr =µcp

Strouhal number: St = LUt0

Non-dimensional energy equation with dimensionless parameters

St∂T ∗

∂t∗+ u∗j

∂T ∗

∂x∗j=

∂T ∗

∂x∗j ∂x∗j

Non-dimensionalization - Summary

Summary of dimensionless Navier-Stokes equations

continuity∂u∗i∂x∗i

momentum∂u∗i∂t∗ + u∗j

∂u∗i∂x∗j

= 1Fr −

∂p∗

∂x∗i+ 1

Re∂2u∗i∂x∗j ∂x∗j

energy St ∂T∗

∂t∗ + u∗j∂T∗

∂x∗j= 1

RePr∂T∗

∂x∗j ∂x∗j

Non-dimensionalization - choosing the parameters

Now how to choose the U, L etc. ?

non dimensionalization

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