The gradient of velocity is generally easy to compute in most CFD post-processing routines. But let’s say you want to find the wall shear stress from this quantity, how would you do that? I’d been searching for an answer to this question and could never really find one (or at least one that was satisfying). Eventually I derived out the following solution and figured I’d post it so that the information was more widely available.

## Initial Definitions #︎

First, let’s define more explicitly what we’re trying to find. The wall shear stress is often given as:

$$\tau_w = \mu \left (\frac{\partial u}{\partial y}\right ) \Bigg\rvert_{y=\text{wall}}$$

However, this isn’t very explicit and really only applies to flat plate boundary layer flows. I’d submit that the “real” definition is dynamic viscosity ($\mu$) times the wall-normal gradient of velocity tangential to the wall taken at the wall, or:

$$\tau_w = \mu \left (\frac{\partial u_\parallel}{\partial n}\right ) \Bigg\rvert_{n=0}$$

This will result in a vector parallel to the wall in the direction of the wall shear stress.

I’ll define the velocity gradient as a tensor $E_{ij}\$:

$$E_{ij} = \frac{\partial u_i}{\partial x_j} = \partial_j u_i$$

Note that $E_{ij}$ is not symmetric and that $\partial_j$ is still an operator with $u_i$ as it’s input, not multiplication.

Lastly, we have the common form of projecting a vector onto a plane given its normal vector:

$$\text{proj}_{\hat n}(\overrightarrow{u}) = \overrightarrow{u} - (\overrightarrow{u} \cdot \hat n) \hat n = u_i - (u_j \hat n_j) \hat n_i$$

where $\hat n$ is the wall-normal unit vector. The right most term is in index summation notation.

## Preamble #︎

### Assumptions #︎

1. We have a wall-normal unit vector $\hat n_i$
2. We have the velocity gradient tensor $E_{ij} = \partial_j u_i$

### Goal #︎

Obtain:

$$\left (\frac{\partial u_\parallel}{\partial n}\right ) \Bigg\rvert_{n=0} = \left (\partial_{\hat n} u_\parallel\right ) \big\rvert_{n=0} = f(E_{ij}, \hat n) = f(\partial_j u_i, \hat n)$$

## Solution #︎

For the impatient, the solution is:

$$\left (\frac{\partial u_\parallel}{\partial n}\right ) \Bigg\rvert_{n=0} = \bigg( \big[(\delta_{ik} - \hat n_k \hat n_i) \hat n_j \big] E_{kj} \bigg) \Bigg\rvert_{n=0} = f(\hat n, E_{ij})$$

The derivation of the above equation is given below.

## Derivation #︎

Notice that the wall shear gradient can be broken into two “terms”:

• gradient in the wall-normal direction
• velocity tangent to the wall

First we’ll define these two “terms” individually

### Gradient in the Wall-Normal Direction #︎

This is simply:

$$\hat n_j \partial_j$$

Gradient in a specific direction should result in a tensor whose rank is the same as it’s input. In other words, the gradient of a scalar in a single direction should result in a scalar (which is a rank 0 tensor). The summation over the $j$ index shows that this is true.

### Velocity Tangent to the Wall #︎

Taking the vector projection formula from Initial Definitions, this is fairly straight forward:

$$u_{i,\parallel} = u_i - (u_k \hat n_k) \hat n_i$$

### Combining Terms #︎

Putting these together, we get:

$$\underbrace{\hat n_j \partial_j}_{\partial_{\hat n}} [\underbrace{u_i - (u_k \hat n_k) \hat n_i }_{u_{\parallel}}]$$

$$\Rightarrow \hat n_j \partial_j \left [u_k (\delta_{ik} - \hat n_k \hat n_i) \right]$$

Using product rule:

$$\Rightarrow (\delta_{ik} - \hat n_k \hat n_i) \hat n_j \partial_j (u_k) + u_k \hat n_j \partial_j (\delta_{ik} - \hat n_k \hat n_i)$$

First, let’s work with the right hand term (RHT):

$$\text{RHT} = u_k \hat n_j \partial_j (\delta_{ik} - \hat n_k \hat n_i)$$

$$\Rightarrow u_k \left [\hat n_j \partial_j (\delta_{ik}) - \hat n_j \partial_j (\hat n_k \hat n_i) \right ]$$

The Kronecker delta is invariant of spacial dimensions, so the left term goes to zero. Then we can do product rule again on the right term.

$$\Rightarrow u_k \left [\hat n_j \cancelto{0}{\partial_j (\delta_{ik})} - \hat n_j \partial_j (\hat n_k \hat n_i)) \right ]$$

$$\Rightarrow -u_k \left [\hat n_i \hat n_j \partial_j (\hat n_k) + \hat n_k \hat n_j \partial_j (\hat n_i) \right ]$$

Here, $\hat n$ is not invariant of spacial location; if you have a non-flat surface, it will change as you move along the wall. However, note that $\hat n_j \partial_j$ is the gradient in the wall-normal direction. The $\hat n$ does not change in the wall-normal direction; it only change in the wall-parallel direction. Thus:

$$\Rightarrow -u_k \left [\hat n_i \cancelto{0}{\hat n_j \partial_j (\hat n_k)} + \hat n_k \cancelto{0}{\hat n_j \partial_j (\hat n_i)} \right ]$$

$$\therefore \text{RHT} = 0$$

Moving back to the original expression, we’re then left with:

$$\partial_{\hat n} u_{i,\parallel} = (\delta_{ik} - \hat n_k \hat n_i) \hat n_j \partial_j (u_k) + \cancelto{0}{u_k \hat n_j \partial_j (\delta_{ik} - \hat n_k \hat n_i)}$$

Note that we already have the gradient of velocity in the last term, thus:

$$\partial_{\hat n} u_{i,\parallel} = (\delta_{ik} - \hat n_k \hat n_i) \hat n_j E_{kj}$$

$$\therefore \left (\frac{\partial u_\parallel}{\partial n}\right ) \Bigg\rvert_{n=0} = \bigg( \big[(\delta_{ik} - \hat n_k \hat n_i) \hat n_j \big] E_{kj} \bigg) \Bigg\rvert_{n=0} = f(\hat n, E_{ij})$$

To obtain $\tau_w$, simply multiply by $\mu$:

$$\tau_w = \mu \bigg( \big[(\delta_{ik} - \hat n_k \hat n_i) \hat n_j \big] E_{kj} \bigg) \Bigg\rvert_{n=0}$$

##### James Wright
###### PhD Student, Aerospace Engineering

I’m interested in CFD, turbulence modeling, motorsports, and disc golf.