Governing equations and boundary conditions for a steady-state compressible viscous flow in an axisymmetric annular orifice

Question

I'm trying to simulate a 2D axisymmetric model of steady-state compressible viscous flow using Mathematica, but I get some errors. There is a chance that I'm making some mistakes with the governing equations and/or the boundary conditions, as suggested in some posts (e.g. here).

Geometry:

The geometry is an axisymmetric tubular step about the x axis forming an annular orifice in 3D.

Assumptions:

1. Axisymmetric about the x axis
2. conduction and radiation are negligable
3. Newtonian fluid with constant viscosity $\eta$
4. Ideal gas (air)
5. Steady-state

Writing the conservation equations in cylindrical form for the axisymmetric problem:

Conservation of mass:

$$ \frac{\partial}{\partial x}\left( \rho v_x \right)+\frac{1}{r}\frac{\partial }{\partial r}\left(r \rho v_r\right)=0 \tag{1}$$

Conservation of linear momentum in axial direction:

$$\frac{\partial}{\partial x}\left( \rho v_x^2+\mathring{R} \rho T \right)+\frac{1}{r}\frac{\partial}{\partial r}\left( r \left( \rho v_r v_x + \eta \frac{\partial v_x}{\partial r} \right)\right) \tag{2}$$

Conservation of linear momentum in radial direction:

$$ \frac{\partial}{\partial x}\left( \rho v_x v_r+\eta \frac{\partial v_r}{\partial x} \right)+ \frac{1}{r}\frac{\partial}{\partial r}\left( r \left( \rho v_r ^2 +\mathring{R} \rho T \right) \right)=0 \tag{3}$$

And energy balance (conservation of heat) more info here:

$$\rho c_v\left( v_x \frac{\partial T}{\partial x} + v_r \frac{\partial T}{\partial r} \right)+ \mathring{R} \rho T \left( \frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right)+ \frac{\partial v_x}{\partial x} \right)+ \eta \left( 2 \left( \frac{\partial v_x}{\partial x} \right)^2+ 2 \left( \frac{\partial v_r}{\partial r} \right)^2+ \left( \frac{\partial v_r}{\partial x}+ \frac{\partial v_x}{\partial r} \right)^2 \\ -\frac{2}{3}\left( \frac{1}{r} \frac{\partial}{\partial r}\left( r v_r \right) + \frac{\partial v_x}{\partial x} \right)^2 \right)=0 \tag{4}$$

The boundary conditions as I presume are:

• (1) Inlet: uniform pressure $P=P_1$
• (2) Outlet: uniform pressure $P=P_0$
• (3) Axis of symmetry $v_r=0$ and $\frac{\partial *}{\partial r}=0$ for all variables $v_r$, $v_x$, $T$ and $\rho$
• (4) no slip walls $v_*=0$

I would appreciate if you could help me know if I'm writing the equations correctly or I'm making any apparent mistakes? Are the boundary conditions enough/correct?

Answer 1

After doing quite some research I think I have the correct equations. Unfortunately all of the equations above (except for the continuity) are wrong.

Linear momentum:

The linear momentum conservation equation in the most compact and general form can be written as:

$$\rho \frac{D \check{v}}{D t}=\check{\nabla} \check{\sigma} \tag{5}$$

Where the $\check{\sigma}$ is the Cauchy stress tensor $\boldsymbol{\sigma}$ in matrix form. The Cauchy stress tensor can be divided into a volumetric/dilatational part and a deviatoric part:

$$\check{\sigma}= \check{\tau} -P\check{I} \tag{6}$$

In a perfect fluid P is the hydrostatic pressure and $\boldsymbol{\tau}$ is the viscous stress tensor.

Constitutive equation:

In an ideal Newtonian fluid viscous stress tensor can be calculated from:

$$ \check{\tau}=\eta\left( \check{\nabla}^T \check{v}+ \left( \check{\nabla}^T \check{v} \right)^T \right) +\lambda \left(\check{\nabla}\check{v}^T\right) \check{I} \tag{7}$$

Where $\eta$ is the absolute/dynamic viscosity and $\lambda$ is the bulk viscosity defined as:

$$\lambda=\kappa-\frac{2}{3}\eta \tag{8}$$

Where $\kappa$ is the dilation/expansion viscosity, which from Stokes’ hypothesis for monatomic gases at low density it is negligible impyling:

$$ \check{\tau}=\eta\left( \check{\nabla}^T \check{v}+ \left( \check{\nabla}^T \check{v} \right)^T -\frac{2}{3} \left(\check{\nabla}.\check{v}^T\right) \check{I} \right) \tag{9}$$

$\eta$ is not necessarily constant. There are some empirical models for example Sutherland:

$$\eta \approx C_S \frac{T^{\left(3/2\right)}}{T+T_S} \tag{10}$$

Where $C_S$ and $T_S$ are constants.

Combining the equations 5, 6 and 9 and also considering the axisymmetry, ideal gas and steady-state assumptions equations of linear momentum in axial and radial directions can be written as:

$$\begin{gather*} \rho \left( v_r \frac{\partial v_r}{\partial r} +v_x \frac{\partial v_r}{\partial x} \right) = \frac{\partial}{\partial r} \left( \eta \left( -\frac{2}{3}\left( \frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right) +\frac{\partial v_x}{\partial x} \right) +2 \frac{\partial v_r}{\partial r} \right) \right) \\ +\frac{\partial}{\partial x}\left( \eta\left( \frac{\partial v_r}{\partial x}+\frac{\partial v_x}{\partial r} \right) \right)+ \frac{2 \eta}{r}\left( \frac{\partial v_r}{\partial r}- \frac{v_r}{r} \right) -\frac{\partial }{\partial r}\left(\mathring{R}\rho T \right) \end{gather*} \tag{11}$$

In radial direction and

$$\begin{gather*} \rho \left( v_r \frac{\partial v_x}{\partial r} +v_x \frac{\partial v_x}{\partial x} \right) = \frac{\partial}{\partial x} \left( \eta \left( -\frac{2}{3}\left( \frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right) +\frac{\partial v_x}{\partial x} \right) +2 \frac{\partial v_x}{\partial x} \right) \right) \\ +\frac{\partial}{\partial r}\left( \eta\left( \frac{\partial v_r}{\partial x}+\frac{\partial v_x}{\partial r} \right) \right)+ \frac{ \eta}{r}\left( \frac{\partial v_r}{\partial x}+ \frac{\partial v_x}{\partial r} \right) -\frac{\partial }{\partial x}\left(\mathring{R}\rho T \right) \end{gather*} \tag{12}$$

In axial direction. (ref1, ref2)

Energy equation:

I'm still not completely sure about the correct form of the energy equation as I have explained here. But assuming my post here is valid then the energy equation in compact form can be written as:

$$ \rho \frac{D e}{D t}=\check{\sigma} : \check{\nabla}^T \check{v} \tag{13}$$

Considering that for an ideal gas $e=c_v T$ (more info here and here) and axisymmetry and steady-state assumptions it expands to:

$$ \begin{gather*} \rho c_v\left( v_x \frac{\partial T}{\partial x} + v_r \frac{\partial T}{\partial r} \right)+ \mathring{R} \rho T \left( \frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right)+ \frac{\partial v_x}{\partial x} \right)= \\ \eta \left( 2 \left( \frac{\partial v_x}{\partial x} \right)^2+ 2 \left( \frac{\partial v_r}{\partial r} \right)^2+ \left( \frac{\partial v_r}{\partial x}+ \frac{\partial v_x}{\partial r} \right)^2 -\frac{2}{3}\left( \frac{1}{r} \frac{\partial}{\partial r}\left( r v_r \right) + \frac{\partial v_x}{\partial x} \right)^2 \right) \end{gather*} \tag{14}$$