No- Slip Condition

No-Slip Condition for Velocity Components at Solid Walls and Its Implementation in Discretized Equations

Boundary Condition in fluid Dynamics
Boundary Condition in CFD

1. No-Slip Condition Explanation:

The no-slip condition is a fundamental boundary condition applied to fluid flow at solid surfaces. It asserts that the velocity of the fluid at the solid boundary is equal to the velocity of the solid surface itself. For most practical cases, this means that at a solid wall, the fluid velocity relative to the wall is zero. Therefore, both the tangential and normal components of the fluid velocity are zero at the wall.

This condition is crucial in fluid dynamics because it reflects the physical reality that the fluid sticks to the solid boundary, resulting in zero relative motion between the fluid and the wall.

2. Implementation in Discretized Equations:

In numerical methods such as finite volume or finite difference methods, the fluid domain is discretized into a grid of control volumes or cells. The no-slip condition is implemented at the boundaries where the fluid is in contact with a solid wall. The key steps to implement this condition in the discretized equations are as follows:

A. For the Normal Velocity Component (v):

  • The normal component of the velocity ( v ) at the wall boundary (say at the grid point ( j = 2 )) is set to zero.

  • The discretized momentum equation for the normal velocity component is then solved for the next grid point in the flow direction (at ( j = 3 )) without any modifications, as the boundary condition directly imposes ( v = 0 ) at the wall.
    B. For the Tangential Velocity Component (u):

  • At the wall boundary (at the grid point closest to the wall, ( j = 2 )), the tangential velocity component ( u ) is also set to zero due to the no-slip condition.

  • In the discretized momentum equation for the tangential velocity component ( u ), the wall force due to shear stress is incorporated as a source term. The wall shear stress is calculated using the velocity gradient near the wall.

    Specifically, the wall shear stress $( \tau_w )$ is given by: $[ \tau_w = \mu \frac{u_P}{\Delta y_P} ]$ where:

    • $( \mu )$ is the dynamic viscosity of the fluid,
    • $( u_P )$ is the velocity at the first grid point ( P ) away from the wall,
    • $( \Delta y_P )$ is the distance of the near-wall node ( P ) from the wall.

    The corresponding shear force $( F_s )$ acting on the wall control volume is then:
    $[ F_s = -\tau_w \times A_{\text{Cell}} ]$
    where $( A_{\text{Cell}} )$ is the area of the control volume face adjacent to the wall.

  • This shear force is included as a source term $( S_P )$ in the discretized u-momentum equation: $[ S_P = -\mu \frac{A_{\text{Cell}}}{\Delta y_P} ]$

C. Pressure Correction:

  • Since the velocity at the wall is known (zero), there is no need to perform a pressure correction for the velocity at the wall.
  • In the discretized pressure correction equation for the cell nearest to the wall, the link to the wall boundary (south) is cut by setting the coefficient $( a_S )$ to zero. The velocity at the wall is thus not altered by pressure corrections during the solution process.

3. Consideration in Turbulent Flow:

  • In turbulent flows, especially near walls, the flow structure is complex due to the presence of a viscous sub-layer, buffer layer, and a logarithmic region. To avoid the high computational cost of resolving all these layers, wall functions are often used. These functions relate the near-wall velocity to the wall shear stress using empirical laws, such as the logarithmic law of the wall.

By applying these methods, the no-slip condition is accurately implemented in the numerical solution, ensuring that the simulation reflects the physical behavior of fluid flow at solid boundaries.