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]:

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

Since (definition of the total derivative):

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

and (conservation of mass) [19]

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

one can write

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

and consequently Equation () amounts to:

\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:

\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]

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

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

\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 () over an element one obtains (using Gauss' theorem):

\frac{\partial }{\partial t } \int_{V}^{} \rho v_i dv + \int_{A}^{} \rho v_i \boldsymbol{v} \cdot \boldsymbol{n} da = \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 ():

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

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