Non-moving Wall

Figure 172: Near-wall stress

At a wall the velocity is zero. However, it is more effective to calculate the stress at the wall directly. The mass conservation amounts to

$$\frac{\partial \rho }{\partial t} + \rho ,_{j} v_j + \rho v_{j,j}=0.$$ (612)

For stationary flow ( $\partial \rho / \partial t =0$) at the wall ( $\boldsymbol{v}=0$) we arrive at

$$v_{j,j}=0$$ (613)

or (Figure 172)

$$\frac{\partial v_n}{\partial n} + \frac{\partial v_t}{\partial t} = 0.$$ (614)

Since $v_t$ does not change along the wall (zero everywhere along the wall) one arrives at

$$\frac{\partial v_n}{\partial n} = 0.$$ (615)

Now, because

$$t_{nn}=2 \mu^T v_{n,n} - \frac{2}{3} (\mu^T v_{k,k} + \rho k)$$ (616)

one obtaines $t_{nn}=0$, since $v_{k,k}=0$ (just derived) and also the turbulent kinetic energy at the wall is zero. For the tangential component one obtains:

$$t_{nt} = \mu^T (v_{n,t}+v_{t,n}).$$ (617)

Since $v_{n,t}=0$, one arrives at

$$t_{nt}=\mu^T v_{t,n}.$$ (618)

The velocity at P (Figure 172) is now decomposed into a component normal and a component tangent to the wall:

$$(\boldsymbol{v}_P)_n = (\boldsymbol{v}_P \cdot \boldsymbol{e}_n) \boldsymbol{e}_n$$ (619)

and

$$(\boldsymbol{v}_P)_t = \boldsymbol{v}_P - (\boldsymbol{v}_P \cdot... ...) \boldsymbol{n} = (\boldsymbol{v}_P \cdot \boldsymbol{e}_t) \boldsymbol{e}_t ,$$ (620)

where $\boldsymbol{e}_n$ and $\boldsymbol{e}_t$ are unit vectors in n- and t-direction, respectively. The stress tensor amounts to:

$$\begin{bmatrix}0 & t_{nt} \\ t_{nt} & t_{tt} \end{bmatrix}$$ (621)

and the normal vector orthogonal and external to the surface satisfies:

$$\boldsymbol{n}= - \boldsymbol{e}_n + 0 \cdot \boldsymbol{e}_t.$$ (622)

This leads to the following stress vector $\boldsymbol{t}$ :

$$\begin{bmatrix}0 & t_{nt} \\ t_{nt} & t_{tt} \end{bmatrix} \cdot \begin{bmatrix}-1 \\ 0 \end{bmatrix} = \begin{bmatrix}0 \\ -t_{nt} \end{bmatrix}$$ (623)

or

$$\boldsymbol{t}= -t_{nt} \boldsymbol{e}_t.$$ (624)

Approximating $v_{t,n}$ by

$$v_{t,n} \approx \frac{\boldsymbol{v}_t \cdot \boldsymbol{e}_t }{(\boldsymbol{r}_S - \boldsymbol{r}_P) \cdot \boldsymbol{n} },$$ (625)

one obtains by combining Equations (618) and (624):

$$\boldsymbol{t}= \frac{-\mu^T \boldsymbol{v}_t \cdot \boldsymbol{e... ... \boldsymbol{n}] }{(\boldsymbol{r}_S - \boldsymbol{r}_P) \cdot \boldsymbol{n}}.$$ (626)

Therefore, the integral at the wall can be approximated by:

$$\left ( \int_{\text{wall}}^{} t_{ij} n_j da \right ) \boldsymbol{... ...}) \boldsymbol{n}}{(\boldsymbol{r}_S - \boldsymbol{r}_P) \cdot \boldsymbol{n}},$$ (627)

where $A_w$ is the area of the wall face. The first term contributes to the left hand side, the second term to the right hand side of the system of equations.