Computational Fluid Dynamics I

Computational Fluid Dynamics I

Exercise 2

1. (a) Derive the vorticity transport equation and the Poisson equation for the stream function Ψ for a two dimensional incompressible and viscous flow.

(b) Formulate the boundary conditions for the stream function and the vorticity component at the boundaries of the channel flow domain shown in the sketch.

∂ ∂

∂

∂ y, v

x, u

u (y) u (y)

v^EE

x=0 v

A A=0 x Wall

Wall

2. Formulate for incompressible flows (without taking into account the energy equation) (a) the Euler equations

- with the velocity vector~v and the pressure p - with stream function Ψ and vorticity componentω (b) the potential equation

- with the velocity componentsu, v (Cauchy–Riemann differential equation) - with Φ

- with Ψ

Determine for a two-dimensional and steady flow the characteristic lines and the type of the equations.

Computational Fluid Dynamics I

Exercise 2 (solution)

1. (a) Navier-Stokes equations 2D, incompressible flow (ρ=const⇒ρt = 0):

u_x+v_y = 0 u_t+uu_x+vu_y+ 1

ρp_x = ν∇²u v_t+uv_x+vv_y +1

ρp_y = ν∇²v

The vorticity transport equation is obtained by taking the curl (∇ ×f~) of the momentum equations: _∂x^∂(y-momentum equation) - _∂y^∂(x-momentum equation)

v_xt+u_xv_x+uv_xx+v_xv_y +vv_xy +¹_ρp_xy

−u_yt−u_yu_x−uu_xy −v_yu_y−vu_yy− ¹_ρp_xy =ν ∇²(v_x

−u_y) where the pressure terms fall out:

(v_x−u_y)_t+u_x(v_x−u_y) +v_y(v_x−u_y)

| {z }

= 0 (mass-conserv. eq.)

+v(v_x−u_y)_y+u(v_x−u_y)_x =ν∇²(v_x−u_y)

With the vorticity componentω =v_x−u_y: ωt+ uωx+vωy

| {z }

convection of vorticity

= ν∇²ω

| {z }

diffusion of vorticity

⇒ Dω

Dt =ν∇²ω

which is the vorticity- (or eddy-) transport equation. The Poisson equation for the stream function Ψ is obtained with u= Ψ_y, v =−Ψ_x:

−ω=−vx+uy = Ψxx+ Ψyy =∇²Ψ

Finally, we have two coupled partial differential equations for the two variablesω and Ψ, the velocitiesuandv in the vorticity-transport equation can be replaced byu= Ψ_y and v =−Ψ_x.

(b) We have boundary conditions given for uand v, but we need them forω and Ψ:

• In- and outflow boundary:

The velocity profile u(y) is given and we know that Ψ_y = ^dΨ_dy =u, therefore integration of dΨ =u(y)dy yields:

Ψ_E(y) = Z y

ywall

u_E(y⁰)dy⁰ + Ψ(y_wall)

where the value at the wall Ψ(y_wall) can be chosen arbitrary as our PDE contains only derivatives of Ψ. For the vorticity boundary condition we compute the derivatives of u and v:

ω_E =v_x−u_y =−∂u_E(y)

∂y

• Solid wall:

The no slip condition u = v = 0 holds, therefore v = Ψ_x = 0 ⇒ Ψ_wall = const.

From the Poisson function for the stream function Ψxx + Ψyy = −ω with Ψ_xx = 0:

⇒ −ω_wall =u_y = Ψ_yy and u_wall = 0 = Ψ_y,wall

Therefore we use a Taylor series expansion for y_wall:

Ψ(y_wall+ ∆y) = Ψ(y_wall) + Ψ_y(y_wall)

| {z }

= 0

∆y+ Ψ_yy(y_wall)∆y² 2 +. . .

⇒Ψ_yy(y_wall) = 2Ψ(ywall+ ∆y)−Ψ(ywall)

∆y²

⇒ω(y_wall) =−Ψ_yy(y_wall) =−2Ψ(y_wall+ ∆y)−Ψ(y_wall)

∆y²

2. (a) Euler equations for incompressible flow (~v, p):

∇ ·~v = 0 D~v

Dt + 1

ρ∇p= 0 characteristic lines (steady 2D flow):

ux+vy = 0 uu_x+vu_y + 1/ρp_x = 0 ⇔

uv_x+vv_y+ 1/ρp_y = 0





∂x ∂y 0

u∂_x+v∂_y 0 ¹_ρ∂_x 0 u∂_x+v∂_y ¹_ρ∂_y







 u v p



= 0

Use chain rule of PDE (u_x =u_ΩΩ_x+u_SS_x) to transform PDE to





Ω_x Ω_y 0

uΩ_x+vΩ_y 0 _ρ¹Ω_x 0 uΩ_x+vΩ_y ¹_ρΩ_y







 u_Ω v_Ω p_Ω





| {z }

crosswise derivative

+





S_x S_y 0

uS_x+vS_y 0 ¹_ρS_x 0 uS_x+vS_y ¹_ρS_y







 u_S v_S p_S



= 0

We need the determinant of the coefficients matrix of the crosswise derivatives to be zero:

Ω_x Ω_y 0

uΩx+vΩy 0 ¹_ρΩx

0 uΩ_x+vΩ_y ¹_ρΩ_y

= 0 =−(uΩ_x+vΩ_y)1

ρΩ²_x−(uΩ_x+vΩ_y)1 ρΩ²_y

⇔(uΩ_x

Ω_y +v)(Ω²_x

Ω²_y + 1) = 0⇒ dy

dx =−Ω_x Ω_y = v

u or Ω_x

Ω_y =±√

−1

i. e. 1 real, 2 imaginary characteristic lines⇒ mixed hyperbolic elliptic type Euler equations (2D) Ψ, ω:

∇²Ψ = −ω Dω

Dt = 0 characteristic lines (steady flow):

Ψxx+ Ψyy =−ω uω_x+vω_y = 0 ⇔

∂xx+∂yy 1 0 u∂_x+v∂_y

Ψ ω

= 0 to solve:

Ω²_x+ Ω²_y 0 0 uΩ_x+vΩ_y

= 0 = (uΩ_x+vΩ_y)(Ω²_x+ Ω²_y)

⇒see Euler equations (~v, p)

(b) Euler equations (incompressible, 2D, irrotational: ω= 0):

(Ψy =u,Ψx=−v,Φx =u,Φy =v)

∇²Φ = Φxx+ Φyy = 0 Potential formulation ~v =∇Φ

∇²Ψ = Ψ_xx+ Ψ_yy = 0 Stream function formulation for which the characteristic slopes are computed by

Q= Ω²_x+ Ω²_y = 0⇒ dy

dx = −Ωx

Ω_y =±√

−1

which results in two imaginary lines ⇒ the PDE is of elliptic type.

Either of the above second-order PDEs can be transformed to a system of two first-order PDEs:

ux+vy = 0 v_x−u_y = 0

which in this case are known as the Cauchy-Riemann differential equation, to compute the characteristic lines solve:

Ω_x Ω_y

−Ω_y Ω_x

= 0⇒Ω²_x+ Ω²_y = 0⇔ dy

dx =−Ω_x

Ω_y =±√

−1

i. e. 2 imaginary characteristic lines ⇒ elliptic type (same results as above)