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31446 Mechanics of fluids: Formula sheet to be used at
written examinations
◮The ǫ − δ identity reads
ǫinmǫmjk = ǫminǫmjk = ǫnmiǫmjk = δij δnk − δik δnj ◮Strain rate tensor, vorticity tensor
∂vi ∂xj
∂vi ∂xj
∂vi ∂xj 2 ∂vi /∂xj
∂vj ∂xi
∂vj ∂xi =
∂vi ∂xj
∂vj ∂xi
∂vi ∂xj
∂vj ∂xi
= Sij + Ωij
◮The vorticity vector is computed as
ω = ∇ × v
ωi = ǫijk
∂vk ∂xj
◮The material derivative
ρ
dΨ dt
= ρ
∂t
∂xj
where Ψ = vi, u, T, k, v i′v j′...
◮The balance equation for mass dρ dt
∂vi ∂xi
◮The balance equation for linear momentum
ρ
dvi dt
∂σji ∂xj
◮The balance equation for internal energy
ρ
du dt
= σji
∂vi ∂xj
∂qi ∂xi ◮The equation for kinetic energy reads
ρ
dk dt
∂viσji ∂xj
− σji
∂vi ∂xj
◮The constitutive law for Newtonian viscous fluids
σij = −pδij + 2μSij −
μSkkδij , σij = −pδij + τij
qi = −k
∂T
∂xi ◮Viscosity
μ: dynamic viscosity
ν: kinematic viscosity (ν = μ/ρ)
◮The continuity equation and the Navier-Stokes equation for incompressible flow with constant viscosity read (conservative form, p denotes the hydrostatic pressure, i.e. p = 0 if vi = 0)
∂vi ∂xi
ρ
∂vi ∂t
∂vivj ∂xj
∂p ∂xi
∂^2 vi ∂xj ∂xj
◮The Navier-Stokes equation for incompressible flow with constant viscosity read (non-conservative form)
ρ
∂vi ∂t
∂vi ∂xj
∂p ∂xi
∂^2 vi ∂xj ∂xj
The viscous stress tensor then reads
τij = 2μSij = μ
∂vi ∂xj
∂vj ∂xi
◮The equation for internal energy reads
ρ du dt
= −p ∂vi ∂xi
μSkkSii Φ
∂xi
k
∂T
∂xi
◮Streamfunction, Ψ; potential, Φ
v 1 =
∂x 2 , v 2 = −
∂x 1
vk =
∂xk ◮The Rayleigh problem
η =
x 2 2
νt
, f =
v 1 V 0
d^2 f dη^2
◮Blasius solution
ξ =
V 1 ,∞
νx 1
x 2 , Ψ = (νV 1 ,∞x 1 )^1 /^2 g
1 2 gg′′^ + g′′′^ = 0
◮The Navier-Stokes (different form of the convective term)
∂vi ∂t
∂k ∂xi
− εijk vj ωk = −
ρ
∂P
∂xi
The linear law: ¯v 1 uτ
uτ x 2 ν
or ¯v+ 1 = x+ 2
The log-law:
¯v 1 uτ
κ
ln
( (^) x 2 uτ ν
κ
ln
x+ 2
+ B
◮The exact k equation reads
∂v¯j k ∂xj
= −v′ iv j′
∂v¯i ∂xj
∂xj
[
ρ
v′ j p′^ +
v j′ v i′v′ i − ν
∂k ∂xj
]
− ν
∂v′ i ∂xj
∂v′ i ∂xj
◮The exact K equation reads
∂¯vj K ∂xj
= ν
∂^2 K
∂xj ∂xj
ρ
∂v¯i ¯p ∂xi
− ν
∂v¯i ∂xj
∂¯vi ∂xj
−
∂v¯iv′ iv j′ ∂xj
∂¯vi ∂xj
◮The modelled k and ε equations
∂k ∂t
∂k ∂xj
= νt
∂¯vi ∂xj
∂v¯j ∂xi
∂¯vi ∂xj
νt σθ
∂ θ¯ ∂xi
−ε +
∂xj
[(
ν +
νt σk
∂k ∂xj
]
∂ε ∂t
∂ε ∂xj
ε k
cε 1 νt
∂v¯i ∂xj
∂¯vj ∂xi
∂¯vi ∂xj
ε k
νt σθ
∂ ¯θ ∂xi
− cε 2
ε^2 k
∂xj
[(
ν +
νt σε
∂ε ∂xj
]
νt = Cμ
k^2 ε ◮Wall functions
uτ =
κ¯v 1 ,P ln(Euτ δx 2 /ν) kP = C− μ 1 /^2 u^2 τ , Cμ = 0. 09
εP = P k^ =
u^3 τ κδx 2 ◮Low-Reynolds number model: different wall boundary conditions for ε:
εwall = ν
∂^2 k ∂x^22
εwall = 2ν
k ∂x 2
εwall =
2 νk x^22
◮In the Boussinesq assumption an eddy (i.e. a turbulent) viscosity is in- troduced to model the unknown Reynolds stresses in Eq. 1. The stresses are
modelled as
v′ iv′ j = −νt
∂v¯i ∂xj
∂¯vj ∂xi
δij k = − 2 νt s¯ij +
δij k
Trick 1:
Ai ∂Bj ∂xk
∂AiBj ∂xk − Bj
∂Ai ∂xk
Trick 2:
Ai ∂Ai ∂xj
∂AiAi ∂xj
