Introduction

A puzzle hunting me when I began to learn CFD is whether my code should be non-dimensionalized. If yes, then how to do this. The advantage of non-dimensionalization lies in the convenience of processing and comparing data. A nondimensional result can represent a series of conditions. But the cost following the advantage is that one should transform the whole equations carefully with suitable scales of quantities. Fortunately, most equations remain the same form after nondimensionalized. But what may confuse readers are the diverse choices of the scales, especially when more complicated physical models are introduced, such as multi-species reaction flow.

Here I summarize the nondimensional process.

Conservation Laws

The motion of compressible viscous fluid is described by a system of equations, that read, in integral, dimensional form:
$V$ being the control volume, $S(V)$ its boundary, $\vec n$ the outward unit normal of $S(V)$ and where $\overrightarrow U $ is the vector of conservative variables, $\overline{\overline F}$ is the conservative fluxes tensor, $\overrightarrow{Q_V}$ is vector of volume source terms and $\overline{\overline{Q_S}}$ is the tensor of surface source terms.
For fluid subjected to external forces and heat sources the above vectors and tensors could be written as:
$$\overrightarrow U = \left[ \begin{array}{*{20}{c}}
\rho \\
\rho \vec v\\
\rho E
\end{array} \right]$$

$$\overline{\overline F} - \overline{\overline {Q_S}} = \left[ \begin{array}{*{20}{c}}
\rho \vec v\\
\rho \vec v\vec v - \overline{\overline \sigma } \\
\rho E\vec v - \overline{\overline \sigma } \cdot \vec v - k\overrightarrow \nabla T
\end{array} \right]$$

$$\overrightarrow {Q_V} =
\left[ \begin{array}{*{20}{c}}
0\\
\rho \overrightarrow {f_e} \\
\rho \overrightarrow {f_e} \cdot \vec v + {q_h}
\end{array} \right]$$
where:

$\rho$ is the density;
$\vec v$ is the velocity vector;
$\vec v \vec v$ is the dyadic product of velocity vector defined as
$\vec u\vec v = \left[ \begin{array}{*{20}{c}}
u_1 v_1 & u_1 v_2 & u_1 v_3 \\
u_2 v_1 & u_2 v_2 & u_2 v_3 \\
u_3 v_1 & u_3 v_2 & u_3 v_3
\end{array} \right]$
$E$ is the specific, total energy;
$T$ is the absolute temperature;
$\overline{\overline \sigma}$ is the total internal stress tensor;
$k$ is the thermal conductivity coefficient, $-k\overrightarrow \nabla T$ being the Fourier’s law of heat conduction; the molecular diffusion and the radiative heat transfer have been neglected;
$\overrightarrow{f_e}$ is the external specific volume forces vector;
$q_h$ is heat sources other than conduction.

The total internal stress could be written as (constitutive law):

$ \overline{\overline \sigma} = -p\overline{\overline I}+ \overline{\overline \tau}$
where:

$p$ is the static, isotropic pressure ($\overline{\overline I}$ is the identity tensor);
$\overline{\overline \tau}$ is the viscous shear stress tensor.

The viscous shear stress tensor, in general case, is defined as:

$\tau_{ij} = \mu \left( \partial_i v_j + \partial_j v_i \right) + \lambda \left( \overrightarrow \nabla \cdot \vec v \right) {\delta_{ij}}$
where:

$\mu$ is the dynamic viscosity of the fluid;
the second viscosity is defined as $\lambda = -\frac{2}{3}\mu$ for a Newtonian fluid in local thermodynamic equilibrium (except very high temperature or pressure ranges); this is the Stokes hypothesis;
$\delta_{ij}$ is the Kronecker delta $\delta_{ij} = \left\{ \begin{array}{*{20}{c}}
0 , \rm{if} \, i \ne j\\
1 , \rm{if} \,i = j \end{array} \right.$.

In tensorial form the shear stress tensor could be written as:

$\overline{\overline \tau } = 2\mu \frac{1}{2}\left( \overrightarrow \nabla \vec v + \overrightarrow \nabla {\vec v}^T \right) + \lambda \left( \overrightarrow \nabla \cdot \vec v \right)\overline{\overline I} $

where $ \frac{1}{2}\left( \overrightarrow \nabla \vec v + \overrightarrow \nabla {\vec v}^T \right)$ is the symmetric part of the velocity vector gradient.

The above system of equation must be completed by the constitutive equation of state. For a perfect gas, i.e. thermally and calorically perfect gas, the equation of state is:

$ p = \rho R T$
where $R$ is the gas constant. This constant is related to the specific heats at constant volume and constant pressure (and their ratio):

$\begin{array}{*{20}{c}}
\gamma = \frac{c_p}{c_v} \\
R = c_p -c_v \\
c_p = \frac{\gamma R}{\gamma -1} \\
c_p = \frac{R}{\gamma -1}
\end{array}$

The other equations of state are:
- $ E = c_v T + \frac{|\vec v|^2}{2} = \frac{p}{\rho(\gamma-1)} + \frac{|\vec v|^2}{2}$ is the total specific energy definition;
- $ a = \sqrt{\frac{\gamma p}{\rho}} $ is the speed of sound (acoustic velocity);
- $ H = c_p T + \frac{|\vec v|^2}{2} = \frac{\gamma p}{\rho(\gamma-1)} + \frac{|\vec v|^2}{2}= \frac{a^2}{\gamma-1} + \frac{|\vec v|^2}{2}$ is the total specific enthalpy definition.

Non-Dimensional Form

In order to derive the non-dimensional form let us introduce non-dimensional quantities $x’=\frac{x}{x_0}$ being $x$ dimensional quantity and $x_0$ an arbitrary reference value. With this nomenclature using a dimensional analysis the above equations could be written as:

$$ \begin{array}{l}
\frac{\rho_0 L_0^3}{t_0}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’ dV’} + {\rho_0}{v_0}L_0^2\oint\limits_{S’(V’)} {\rho’ \overrightarrow {v’} \cdot \vec ndS’} = 0 \\
\frac{\rho_0 v_0 L_0^3}{t_0}\frac{\partial}{\partial t’}\int\limits_{V’} \rho’\overrightarrow{v’} dV’ + \rho_0 v_0^2 L_0^2 \oint \limits_{S’(V’)} \rho’\overrightarrow {v’} \overrightarrow {v’} \cdot \vec ndS’ + p_0 L_0^2\oint\limits_{S’(V’)} p’\overline{\overline I} \cdot \vec ndS’ - \tau_0 L_0^2\oint\limits_{S’(V’)} \overline{\overline {\tau’}} \cdot \vec ndS’ = \rho_0 f_0 L_0^3\int\limits_{V’} \rho’ \overrightarrow{f’_e} dV’ \\
\frac{\rho _0 E_0 L_0^3}{t_0}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’E’dV’} + {\rho _0}{E_0}{v_0}L_0^2\oint\limits_{S’(V’)} \rho’E’\overrightarrow {v’} \cdot \vec ndS’ + {p_0}L_0^2\oint\limits_{S’(V’)} p’\overrightarrow {v’} \cdot \vec ndS’ - {\tau _0}{v_0}L_0^2\oint\limits_{S’(V’)} \left( \overline{\overline {\tau’}} \cdot \overrightarrow {v’} \right) \cdot \vec ndS’ + \\
-\frac{k_0 T_0 L_0^2}{L_0}\oint\limits_{S’(V’)} k’\overrightarrow {\nabla’} T’ \cdot \vec ndS’ = {\rho _0}{f_0}{v_0}L_0^3\int\limits_{V’} {\rho’\overrightarrow {f’_e} \cdot \overrightarrow {v’} dV’} + {\rho _0}{q_0}L_0^3\int\limits_{V’} {\rho’ {q’}_h dV’}
\end{array} $$

In order to make non-dimensional the above system divide for ${\rho _0}{v_0}L_0^2$, ${\rho _0}{v_0^2}L_0^2$ and ${\rho _0}{E_0}{v_0}L_0^2$ the conservation equation of mass, momentum and energy respectively, obtaining:

$$ \begin{array}{l}
\frac{L_0}{v_0 t_0}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’dV’} + \oint\limits_{S’(V’)} {\rho’\overrightarrow {v’} \cdot \vec ndS’} = 0\\
\frac{L_0}{v_0 t_0}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’\overrightarrow {v’} dV’} + \oint\limits_{S’(V’)} {\rho’\overrightarrow {v’} \overrightarrow {v’} \cdot \vec ndS’} + \frac{p_0}{\rho_0 v_0^2}\oint\limits_{S’(V’)} {p’\overline{\overline I} \cdot \vec ndS’}- \frac{\tau_0}{\rho_0 v_0^2}\oint\limits_{S’(V’)} {\overline{\overline {\tau’}} \cdot \vec ndS’} = \frac{f_0 L_0}{v_0^2}\int\limits_{V’} \rho’\overrightarrow {f’_e} dV’ \
\frac{L_0}{v_0 t_0}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’E’dV’} + \oint\limits_{S’(V’)} {\rho’E’\overrightarrow {v’} \cdot \vec ndS’} + \frac{p_0}{\rho_0 E_0 v_0}\oint\limits_{S’(V’)} {p’\overrightarrow {v’} \cdot \vec ndS’} - \frac{\tau_0}{\rho_0 E_0}\oint\limits_{S’(V’)} {\left( \overline{\overline {\tau’}} \cdot \overrightarrow {v’} \right) \cdot \vec ndS’} + \ - \frac{k_0 T_0}{\rho_0 E_0v_0 L_0}\oint\limits_{S’(V’)} {k’\overrightarrow {\nabla’} T’ \cdot \vec ndS’} = \frac{f_0 L_0}{E_0}\int\limits_{V’} {\rho’\overrightarrow {f’_e} \cdot \overrightarrow {v’} dV’} + \frac{q_0 L_0}{E_0 v_0}\int\limits_{V’} {\rho’ q’_h dV’}
\end{array}$$

It is possible to recognize 6 non-dimensional numbers:

$ \rm{St} = \frac{L_0}{v_0 t_0}$ Strouhal number; it is the ratio between characteristic frequency and the fluid dynamic one; for non periodic flow it is set to 1;
$ \rm{Ru} = \frac{\rho_0 v_0^2}{p_0}$ Ruark number; it is the ratio between inertial (convective) force and pressure one; for a Newtonian fluid, according to the equation of state $p_0=\rho_0 R T_0$, we have $Ru=\gamma Ma^2$ ;
$ \rm{Ma} = \frac{v_0}{a_0}$ Mach number; it is the ratio between velocity and speed of sound;
$ \rm{Re} = \frac{\rho_0 v_0 L_0}{\mu_0}$ Reynolds number; it is the ratio between inertial (convective) force and viscous one;
$ \rm{Fr} = \sqrt{\frac{v_0^2}{f_0 L_0}}$ Froude number; it is the ratio between inertial (convective) force and volume (mass) one;
$ \rm{Pr} = \frac{\mu_0 c_{p0}}{k_0}$ Prandtl number; it is the ratio between momentum diffusion and heat one;
$ \rm{Ec} = \frac{v_0^2}{c_{p0}T_0}$ Eckert number; it is the ratio between kinetic energy and enthalpy one;

By means of the above non-dimensional numbers the conservation equations could be written as:

$$ \begin{array}{l}
\rm{St}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’dV’} + \oint\limits_{S’(V’)} {\rho’\overrightarrow {v’} \cdot \vec ndS’} = 0\\
\rm{St}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’\overrightarrow {v’} dV’} + \oint\limits_{S’(V’)} {\rho’\overrightarrow {v’} \overrightarrow {v’} \cdot \vec ndS’} + \frac{1}{\rm{Ru}}\oint\limits_{S’(V’)} {p’\overline{\overline I} \cdot \vec ndS’} - \frac{1}{\rm{Re}}\oint\limits_{S’(V’)} {\overline{\overline {\tau’}} \cdot \vec ndS’} = \frac{1}{\rm{Fr}^2}\int\limits_{V’} {\rho’\overrightarrow {f’_e} dV’} \\
\rm{St}\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’E’dV’} + \oint\limits_{S’(V’)} {\rho’E’\overrightarrow {v’} \cdot \vec ndS’} + \frac{\rm{Ec}}{\rm{Ru}}\oint\limits_{S’(V’)} {p’\overrightarrow {v’} \cdot \vec ndS’} - \frac{\rm{Ec}}{\rm{Re}}\oint\limits_{S’(V’)} {\left( {\overline{\overline {\tau’}} \cdot \overrightarrow {v’} } \right) \cdot \vec ndS’} + \ - \frac{1}{\rm{PrRe}}\oint\limits_{S’(V’)} {k’\overrightarrow {\nabla’} T’ \cdot \vec ndS’} = \frac{\rm{Ec}}{\rm{Fr}^2}\int\limits_{V’} {\rho’\overrightarrow {f’_e} \cdot \overrightarrow {v’} dV’} + \frac{q_0 L_0}{E_0 v_0}\int\limits_{V’} {\rho’ q’_h dV’}
\end{array} $$

We are interested in Newtonian fluid with no volume source terms in which $\rm{St} = 1$, $\rm{Ru} = 1 => \rm{Ma} = \frac{1}{\sqrt{\gamma}}$ and $\rm{Ec} = 1$, $\overrightarrow{Q_V}=\overrightarrow{0}$:

$ t_0 = \frac{L_0}{v_0}$
$ p_0 = \rho_0 v_0^2 => a_0 = \sqrt{\gamma}v_0$
$ c_{p0}T_0 = v_0^2$

As a consequence the non dimensional conservation equations are:

$$\begin{array}{l}
\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’dV’} + \oint\limits_{S’(V’)} {\rho’\overrightarrow {v’} \cdot \vec ndS’} = 0\\
\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’\overrightarrow {v’} dV’} + \oint\limits_{S’(V’)} {( \rho’\overrightarrow {v’} \overrightarrow {v’} + p’\overline{\overline I} ) \cdot \vec ndS’} - \frac{1}{\rm{Re}}\oint\limits_{S’(V’)} {\overline{\overline {\tau’}} \cdot \vec ndS’} = 0\\
\frac{\partial }{\partial t’}\int\limits_{V’} {\rho’E’dV’} + \oint\limits_{S’(V’)} {(\rho’E’ + p’)\overrightarrow {v’} \cdot \vec ndS’} - \frac{1}{\rm{Re}}\oint\limits_{S’(V’)} {( {\overline{\overline {\tau’}} \cdot \overrightarrow {v’} }) \cdot \vec ndS’} - \frac{1}{\rm{PrRe}}\oint\limits_{S’(V’)} {k’\overrightarrow {\nabla’} T’ \cdot \vec ndS’} = 0
\end{array} $$

The selected non-dimensional numbers are:

$ \rm{Re} = \frac{\rho_0 v_0 L_0}{\mu_0}$ Reynolds number;
$ \rm{Pr} = \frac{\mu_0 c_{p0}}{k_0}$ Prandtl number;

The above non-dimensional numbers are dependent each other. In Reynolds number $L_0$ and $v_0$ are present. Similarly in Reynolds and Prandtl numbers $\mu_0$ is present. Therefore fixing the above non-dimensional numbers the following reference values must be fixed:

$ L_0 $ reference length;
$ \rho_0 $ reference density;
$ v_0 $ reference velocity;
$ \mu_0 $ reference dynamic viscosity;
$ c_0 $ reference specific heat;
$ k_0 $ reference thermal conductivity coefficient.

We have fixed the following reference values:

$ L_0 $ reference length;
$ \rho_0 $ reference density;
$ v_0 $ reference velocity;
$ c_0 $ reference specific heat.

These selections with the above selected non-dimensional numbers fix all other reference quantities:

$ \mu_0 = \frac{\rho_0 v_0 L_0}{\rm{Re}}$ reference dynamic viscosity;
$ k_0 = \frac{\mu_0 c_0}{\rm{Pr}}$ reference thermal conductivity coefficient.
$ t_0 = \frac{L_0}{v_0}$ reference time;
$ p_0 = \rho_0 v_0^2 $ reference pressure;
$ a_0 = \sqrt{\gamma}v_0 $ reference speed of sound;
$ T_0 = \frac{v_0^2}{c_0} $ reference temperature;
$ E_0 = v_0^2 $ reference specific energy;
$ q_0 = \frac{v_0^3}{L_0} $ reference specific heat.

The above equations constitute a system of partial differential equations closed by enforcing appropriate boundary conditions and initial conditions (at physical and computational boundaries).

参考资料