Non-moving Wall

Figure: 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 )

\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 ) 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{e}_n) \boldsymbol{e}_n =\boldsymbol{v}_P - (\boldsymbol{v}_P \cdot \boldsymbol{n}) \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 () and ():

\boldsymbol{t}= \frac{-\mu^T \boldsymbol{v}_t \cdot \boldsymbol{e}_t }{(\boldsymbol{r}_S - \boldsymbol{r}_P) \cdot \boldsymbol{n}} \cdot \boldsymbol{e}_t = - \frac{\mu^T [\boldsymbol{v}_P - ( \boldsymbol{v}_P \cdot \boldsymbol{n}) \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{e}_i \approx - \frac{\mu^{T(m-1)} A_w}{(\boldsymbol{r}_S - \boldsymbol{r}_P) \cdot \boldsymbol{n}} \boldsymbol{v}_P^{(m)} + \mu^{T(m-1)} A_w \frac{(\boldsymbol{v}_P^{(m-1)} \cdot \boldsymbol{n}) \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.