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Contents

• Navier-Stokes
• Linearized Euler (1D)
  • Perturbations
• Linearized Navier-Stokes
• Acoustics Equations
  • Generalized Equation
• Other Equations

Navier-Stokes

NS equations are a set of non-linear partial derivative equations of conservation of mass, momentum, and energy:

$$\begin{aligned} \frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mathbf v)&=0\\ \frac{\partial(\rho\mathbf v)}{\partial t}+\nabla\cdot \left( \rho\mathbf v\mathbf v + p\mathbf I \right)&=\nabla\cdot\mathbf\tau+\mathbf f_m\\ \frac{\partial(\rho E)}{\partial t}+\nabla\cdot(\rho\mathbf vH)&=\nabla\cdot(k\mathbf\nabla T)+\nabla\cdot(\mathbf\tau\cdot\mathbf v)+\mathbf f_m\cdot\mathbf v \end{aligned}$$

Equations of state:

• $p=\rho R_gT$
• $e=c_vT$
  • $R_g=c_p-c_v$
• $E=e+V^2/2$
• $H=E+p/\rho$

These equations are even more complicated in CFD as one needs to add artificial viscosity and equations for modeling turbulences.

Some notes:

• One should use conservative variables to solve discontinuities correctly
• Convective terms are more completed than viscous terms

Linearized Euler (1D)

Approximation of ideal gas, no viscosity and heat transfer ($\mathbf\tau'=\mathbf q=0$, with $\mathbf\tau=-p\mathbf I+\mathbf\tau'$):

$$\frac{\partial}{\partial t} \begin{bmatrix} \rho\\ \rho u\\ \rho E \end{bmatrix}+\frac{\partial}{\partial x} \begin{bmatrix} \rho u\\ \rho u^2+p\\ \rho u H \end{bmatrix}= \begin{bmatrix} 0\\0\\0 \end{bmatrix}$$

Defining conservative variables:

• $u_1=\rho$
• $u_2=\rho u$
• $u_3=\rho E$

And with the vector of conservative variable: $\mathbf U=(u_1,u_2,u_3)^T$

$$\frac{\partial\mathbf U}{\partial t}+\frac{\partial\mathbf F}{\partial x}=\mathbf 0$$

Where $\mathbf F$ is the vector of conservative flux:

$$\begin{aligned} f_1&=u_2\\ f_2&=\frac{u_2^2}{u_1}+(\gamma-1)u_3-\frac{1}{2}(\gamma-1)\frac{u_2^2}{u_1}\\ f_3&=u_2\left( \gamma\frac{u_3}{u_1}-\frac{1}{2}(\gamma-1)\frac{u_2^2}{u_1^2} \right) \end{aligned}$$

In quasi-linear form:

$$\boxed{\frac{\partial\mathbf U}{\partial t}+\frac{\partial\mathbf F}{\partial\mathbf U}\frac{\partial \mathbf U}{\partial x}=\mathbf 0}$$

And by defining $A=\partial\mathbf F/\partial\mathbf U$:

$$A=\frac{\partial f_i}{\partial u_j}= \begin{bmatrix} 0 & 1 & 0 \\ \dfrac{1}{2}(\gamma-3)\left( \dfrac{u_2}{u_1} \right)^2 & (3-\gamma)\dfrac{u_2}{u_1} & \gamma-1 \\ -\gamma\dfrac{u_2u_3}{u_1^2}+(\gamma-1)\left( \dfrac{u_2}{u_3} \right)^3 & \gamma\dfrac{u_3}{u_1}-\dfrac{3}{2}(\gamma-1)\left( \dfrac{u_2}{u_1} \right)^2 & \gamma\dfrac{u_2}{u_1} \end{bmatrix}$$

Using conservative variables and $a=\sqrt{\gamma R_g T}$:

$$A= \begin{bmatrix} 0 & 1 & 0 \\ \dfrac{1}{2}(\gamma-3)u^2 & (3-\gamma)u & \gamma-1 \\ \dfrac{1}{2}(\gamma-2)u^3-\dfrac{a^2u}{\gamma-1} & \dfrac{3-2\gamma}{2}u^2+\dfrac{a^2}{\gamma-1} & \gamma u \end{bmatrix}$$

Perturbations

Supposing $\mathbf U(x,t)=\mathbf U_0+\mathbf u'(x,t)$ with $||\mathbf u'||\ll||\mathbf U_0||$ where:

• $\mathbf U_0$: Uniform and stationary solution
• $\mathbf u'(x,t)$: General perturbation

Using Taylor expansion with the quasi-linear form, retaining only 1st order components:

$$\frac{\partial\mathbf u'}{\partial t}+A(\mathbf U_0)\frac{\partial\mathbf u'}{\partial x}=\mathbf 0$$

This equation is a hyperbolic system if the eigenvalues for $A(\mathbf U_0)$ are reals: $A(\mathbf U_0)\mathbf u_i^R=\lambda_i\mathbf u_i^R$

• $\lambda_1=u_0$
• $\lambda_2=u_0+a_0$
• $\lambda_2=u_0-a_0$

With $a_0=\sqrt{\gamma R_gT_0}=\sqrt{\gamma p_0/\rho_0}$, and right eigenvectors:

$$\mathbf u_1^R= \begin{bmatrix} 1\\u_0\\\dfrac{u_0^2}{2} \end{bmatrix}\quad;\quad \mathbf u_2^R= \begin{bmatrix} 1\\ u_0+a_0\\ \dfrac{u_0^2}{2}+\dfrac{a_0^2}{\gamma-1}+a_0u_0 \end{bmatrix}\quad;\quad \mathbf u_3^R= \begin{bmatrix} 1\\ u_0-a_0\\ \dfrac{u_0^2}{2}+\dfrac{a_0^2}{\gamma-1}-a_0u_0 \end{bmatrix}$$

Defining $T_0=(\mathbf u_1^R,\mathbf u_2^R,\mathbf u_3^R)$ and $\mathbf w$:

$$T_0^{-1}A_0T_0=\Lambda_0= \begin{bmatrix} \lambda_1&0&0\\ 0&\lambda_2&0\\ 0&0&\lambda_3 \end{bmatrix}\quad;\quad \mathbf w=T_0^{-1}\mathbf u'$$

Adding to the linear equation:

$$\frac{\partial\mathbf w}{\partial t}+\Lambda_0\frac{\partial\mathbf w}{\partial x}=\mathbf 0$$

where:

• $w_1=p'-a_0^2\rho'$
• $w_2=p'+\rho_0a_0u'$
• $w_3=p'-\rho_0a_0u'$

Boundary conditions:

• $M_0<1$:
  • 2 conditions in $x=0$ for downstream waves (entropy and acoustic)
  • 1 condition in $x=L$ for upstream wave
• $M_0>1$:
  • 3 conditions in $x=0$ for all three waves

Linearized Navier-Stokes

$$\frac{\partial}{\partial t} \begin{bmatrix} \rho\\ \rho u\\ \rho E \end{bmatrix}+\frac{\partial}{\partial x} \begin{bmatrix} \rho u\\ \rho u^2+p\\ \rho u H \end{bmatrix}= \frac{\partial}{\partial x} \begin{bmatrix} 0\\\mu_x\\\mu uu_x+\kappa T_x \end{bmatrix}$$

Compact form:

$$\frac{\partial\mathbf U}{\partial t}+\frac{\partial\mathbf f(\mathbf U)}{\partial x}=\frac{\partial\mathbf f_v(\mathbf U)}{\partial x}$$

• $\mathbf f$: Convective flux
• $\mathbf f_v$: Viscous flux

For analytical study, better user primitive variables: $\mathbf V=(\rho,u,p)^T$. Therefore:

$$\frac{\partial}{\partial t} \begin{bmatrix} \rho\\u\\p \end{bmatrix}+ \begin{bmatrix} u&\rho&0\\ 0&u&1/\rho\\ 0&\gamma p&u \end{bmatrix} \frac{\partial}{\partial x} \begin{bmatrix} \rho\\u\\p \end{bmatrix}= \begin{bmatrix} 0\\ (\mu u_x)_x\\ (\gamma-1)((kT_x)_x+\mu u_x^2) \end{bmatrix}$$

By linearizing the system:

$$\frac{\partial}{\partial t} \begin{bmatrix} \rho'\\u'\\p' \end{bmatrix}+ \underbrace{ \begin{bmatrix} u_0&\rho_0&0\\ 0&u_0&1/\rho_0\\ 0&\gamma p_0&u_0 \end{bmatrix}}_{A_0} \frac{\partial}{\partial x} \begin{bmatrix} \rho'\\u'\\p' \end{bmatrix}= \begin{bmatrix} 0\\ \mu u_{xx}'\\ \dfrac{(\gamma-1)k_0}{\rho_0R_g} \left( p'_{xx}-\dfrac{p_0}{\rho_0}\rho'_{xx} \right) \end{bmatrix}$$

Eigenvalues for $A_0$ are the same, but the eigenvectors are now the columns of $Q_0$:

$$Q_0= \begin{bmatrix} 1&\dfrac{\rho_0}{2 a_0}&-\dfrac{\rho_0}{2a_0}\\ 0&\dfrac{1}{2}&\dfrac{1}{2}\\ 0&\dfrac{\rho_0a_0}{2}&-\dfrac{\rho_0a_0}{a} \end{bmatrix}\quad;\quad Q_0^{-1}= \begin{bmatrix} 1&0&-\dfrac{1}{a_0^2}\\ 0&1&\dfrac{1}{\rho_0a_0}\\ 0&1&-\dfrac{1}{\rho_0a_0} \end{bmatrix}$$

By multiplying $Q_0^{-1}$ to the linearized equation:

$$\frac{\partial}{\partial t} \begin{bmatrix} w_1\\w_2\\w_3 \end{bmatrix}+ \begin{bmatrix} u_0&0&0\\ 0&u_0+a_0&0\\ 0&0&u_0-a_0 \end{bmatrix} \frac{\partial}{\partial x} \begin{bmatrix} w_1\\w_2\\w_3 \end{bmatrix}= B\frac{\partial^2}{\partial x^2} \begin{bmatrix} w_1\\w_2\\w_3 \end{bmatrix}$$

Stripping dimensions with the base field values:

• $\overline u=u'/a_0$
• $\overline p=p'/(\rho_0a_0^2)$
• $\overline t=t/(L_c/a_0)$
• $\overline x=x/L_c$

The equation is therefore:

$$\begin{gathered} \frac{\partial}{\partial t} \begin{bmatrix} \overline w_1\\\overline w_2\\\overline w_3 \end{bmatrix}+ \begin{bmatrix} \overline u_0&0&0\\ 0&\overline u_0+1&0\\ 0&0&\overline u_0-1 \end{bmatrix} \frac{\partial}{\partial x} \begin{bmatrix} \overline w_1\\\overline w_2\\\overline w_3 \end{bmatrix}= \overline B\frac{\partial^2}{\partial x^2} \begin{bmatrix} \overline w'_1\\\overline w'_2\\\overline w'_3 \end{bmatrix}\\ B=\frac{1}{\mathrm{Re}} \begin{bmatrix} 0&0&0\\ 0&1/2&-1/2&\\ 0&-1/2&1/2 \end{bmatrix}+\frac{1}{\mathrm{Re\,Pr}} \begin{bmatrix} 1&\dfrac{\gamma-1}{2}&\dfrac{\gamma-1}{2}\\ 1&\dfrac{\gamma}{2}&\dfrac{\gamma-2}{2}\\ 1&\dfrac{\gamma-2}{2}&\dfrac{\gamma}{2} \end{bmatrix} \end{gathered}$$

• $\mathrm{Re}=\rho_0u_0L_c/\mu_0$
• $\mathrm{Pr}=c_p\mu_0/\kappa_0$
• $\overline w_1=\overline p'-\overline\rho'$
• $\overline w_1=\overline p'+\overline u'$
• $\overline w_1=\overline p'-\overline u'$

Some conclusions:

• If $\kappa_0=0$: Entropy equation decouples, and $s'$ is conserved along the path lines
• If $\mathrm{Re}\gg1$: Waves decouple. Perturbed characteristic variables are conserved along the path lines, slightly damped due to the viscosity

Entropy equation:

$$\rho T\frac{\DD s}{\DD t}=\mathbf\nabla\cdot(\kappa\mathbf\nabla T)+\mathbf\Phi_v+Q$$

Linearized entropy equation:

$$\rho_0T_0(\partial_ts'+u_0\partial_xs')=\kappa_0\partial_x^2T'+\cancel{\mathbf\Phi'_v}$$

Vorticity equation ($\mathbf\omega=\mathbf\nabla\times\mathbf u$) for isentropic and ideal flow, without forces from masses:

$$\frac{\DD (\mathbf\omega/\rho)}{\DD t}=\frac{\mathbf\omega}{\rho}\cdot\mathbf\nabla\mathbf V$$

Acoustics Equations

Using the following decomposition in the Euler’s equation:

$$\begin{aligned} u&=u'\\ \rho&=\rho_0+\rho'\\ p&=p_0+p' \end{aligned}$$

We can obtain:

$$\begin{aligned} &0=\partial_t(\rho_0+\rho')+\partial_x[(\rho_0+\rho')u']\\ &0=\partial_t[(\rho_0+\rho')u']+\partial_x[(\rho_0+\rho')u'u'+p_0+p'] \end{aligned}$$

As $\rho_0$ and $p_0$ (with $u_0=0$) also verifies Euler’s equation:

$$\partial_t\rho_0=0\quad;\quad\partial_xp_0=0$$

Using these values we can simplify the equation above:

$$\begin{aligned} &0=\partial_t\rho'+\partial_x[(\rho_0+\rho')u']\\ &0=\rho_0\partial_tu'+\partial_t(\rho'u')+\partial_x[(\rho_0+\rho')u'u']+\partial_xp' \end{aligned}$$

Looking for linear solutions, neglecting quadratic terms of the perturbations:

$$\begin{aligned} 0&=\partial_t\rho'+\partial_x(\rho_0u')\\ 0&=\rho_0\partial_tu'+\partial_x\rho' \end{aligned}$$

The relationship between $p'$ and $\rho'$ is:

$$\begin{rcases} p&=p_0+p'\\ p(\rho)&\approx p(\rho_0)+ \left. \dfrac{\partial p}{\partial\rho} \right|_{\rho_0}\rho' \end{rcases}p'=\left. \frac{\partial p}{\partial \rho} \right|_{s_0,\rho_0}\rho'=a_0^2\rho'$$

Using this property results in:

$$\begin{aligned} 0&=\frac{1}{a_0^2}\partial_tp'+\rho_0\partial_xu'\\ 0&=\rho_0\partial_t u'+\partial_xp' \end{aligned}$$

Which is a system of two linear equations with two unknowns.

Deriving the first equation with respect to time and the second with respect to $x$:

$$\begin{rcases} \dfrac{1}{a_0^2}\partial_{tt}p'+\rho_0\partial_{tx}(u')&=0\\ \rho_0\partial_{xt}u'+\partial_{xx}p'&=0 \end{rcases} \frac{1}{a_0^2}\partial_{tt}p'=\partial_{xx}p'$$

Which is the equation of the perturbation in the pressure field. In 2D:

$$\frac{1}{a_0^2}\partial_{tt}p'=\nabla^2p'\quad;\quad\nabla^2=\partial_{xx}+\partial_{yy}$$

Generalized Equation

Considering the acoustic equation with base flux speed $u_0\ne0$, the linearized Euler’s equations in 1D:

$$\begin{aligned} 0&=\frac{1}{a_0^2}\frac{\partial p'}{\partial t}+\rho_0\frac{\partial u'}{\partial x}+\frac{u_0}{a_0^2}\frac{\partial p'}{\partial x}\\ 0&=\frac{\partial u'}{\partial t}+\frac{1}{\rho_0}\frac{\partial p'}{\partial x}+u_x\frac{\partial u'}{\partial x} \end{aligned}$$

Combining both:

$$\begin{gathered} \frac{1}{a_0^2}\frac{\DD_0^2p'}{\DD t^2}=\nabla^2p'\quad;\quad\frac{\DD_0}{\DD t}=\frac{\partial}{\partial t}+u_0\nabla\\ \frac{1}{a_0^2}\left( \frac{\partial^2p'}{\partial t^2}+2u_0\frac{\partial^2p'}{\partial t\partial x}+u_0^2\frac{\partial^2p'}{\partial x^2} \right)=\frac{\partial^2p'}{\partial x^2} \end{gathered}$$

which transforms into an equation with a 2nd temporal derivative with respect to time into two of 1st order:

$$\frac{\partial}{\partial t} \begin{pmatrix} p_1\\p_2 \end{pmatrix}= \begin{pmatrix} 0&1\\ (a_0^2-u_0^2)(\partial^2/\partial x^2)&-2u_0(\partial/\partial x) \end{pmatrix} \begin{pmatrix} p_1\\p_2 \end{pmatrix}$$

where $p_1=p'$ and $p_2=\dot p'$

Other Equations

• Linear waves: $\partial_tu+a\partial_xu=0$
• Viscous lineal waves: $\partial_tu+a\partial_xu=\nu\partial_x^2u$
• Burgers: $\partial_tu+u\partial_xu=0$
• Viscous burgers (Convection-Diffusion): $\partial_tu+u\partial_xu=\nu\partial_x^2u$
• Heat: $\partial_tu=\alpha\partial_x^2u$
• Stationary heat (Poisson): $\partial_x^2u=0$
• Linear wave with source: $\partial_tu+a\partial_xu=\alpha u$
• 2D linear waves: $\partial_tu+A\partial_xu+B\partial_yu=0$