Conservation of momentum (compressible flow)

The conservation principles are of utmost importance in fluid dynamics. They lead to sets of linear equations the solution of which yields the fields we are looking for (velocity, pressure, temperature...). The conservation of momentum can be written in the following component form (in spatial carthesian coordinates) [19]:

$\displaystyle \rho \frac{D v_i}{Dt} = \sigma_{ij,j} + \rho f_i .$ (585)

Since (definition of the total derivative):

$\displaystyle \rho \frac{Dv_i}{Dt} = \rho \frac{\partial v_i}{\partial t} + \rho v_{i,j}v_j$ (586)

and (conservation of mass) [19]

$\displaystyle \frac{\partial \rho }{\partial t} + (\rho v_j),_{j} =0,$ (587)

one can write

$\displaystyle \rho \frac{Dv_i}{Dt} = \frac{\partial \rho v_i}{\partial t}+ (\rho v_i v_j),_{j}$ (588)

and consequently Equation (585) amounts to:

$\displaystyle \frac{\partial \rho v_i}{\partial t} + (\rho v_i v_j),_{j} = t_{ij,j} - p,_{j} \delta_{ij} + \rho f_i,$ (589)

since the Cauchy stress $ \sigma_{ij}$ can be written as the viscous stress $ t_{ij}$ minus the hydrostatic pressure $ p$:

$\displaystyle \sigma_{ij}=t_{ij} - p \delta_{ij}.$ (590)

The viscous stress $ t_{ij}$ can be written as the sum of the laminar viscous stress $ t_{ij}^l$ and the turbulent viscous stress $ t_{ij}^t$ satisfying [67]

$\displaystyle t_{ij}^l = \mu (v_{i,j}+v_{j,i}-\frac{2}{3} v_{k,k} \delta_{ij})$ (591)

$\displaystyle t_{ij}^t = \mu_t (v_{i,j}+v_{j,i}-\frac{2}{3} v_{k,k} \delta_{ij})-\frac{2}{3} \rho k \delta_{ij},$ (592)

($ k$ is the turbulent kinetic energy and $ \mu_t$ is the turbulent viscosity) leading to

$\displaystyle \sigma_{ij}=(\mu + \mu_t) (v_{i,j}+v_{j,i}-\frac{2}{3} v_{k,k}\delta_{ij})-(p+\frac{2}{3}\rho k) \delta_{ij}.$ (593)

Integrating Equation (589) over an element one obtains (using Gauss' theorem):

$\displaystyle \frac{\partial }{\partial t } \int_{V}^{} \rho v_i dv + \int_{A}^...
...a = \int_{A}^{} t_{ij} n_j da - \int_{A}^{} p n_i da + \int_{V}^{} \rho f_i dv,$ (594)

where $ V$ is the volume of the element and $ A$ the external surface (which is the sum of the area of all external faces of the element). The area of a face is calculated by considering it as a 2-dimensional finite element and calculating the Jacobian vector at the center (1-point integration). The volume is obtained by replacing $ \phi$ by the coordinate $ x$ and $ k$ by $ 1$ in Equation (559):

$\displaystyle V_p=\sum_{f}^{} x_f (n_1)_f A_f.$ (595)

Now, turning to Equation (594) each term is considered in detail for element P and iteration $ (m)$ of increment $ n$.



Subsections