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Deriving the Turbulent Shear Stress Equation

Key Takeaways

  • To create quality designs, it is vital to understand the velocity fluctuations in a turbulent flow.

  • In turbulent flow, shear stress is a macroscopic phenomenon resulting from the momentum transfer between finite-sized fluid particles that move randomly. 

  • In the pipe-boundary layer flow, shear stress is dominant, which is associated with the y-direction normal to the wall.

  Turbulent flow

 A turbulent flow is an irregular flow where the flow parameters vary randomly with time and space coordinates.

In engineering, most of the flows that we deal with are turbulent. A turbulent flow is an irregular flow where the flow parameters vary randomly with time and space coordinates. Random fluctuations in flow parameters produce statistically distinct average values. The shear stress experienced in turbulent flow also fluctuates, and the variation is different for different layers in the flow. The same applies to the velocity profile of turbulent flow.

To create quality designs, it is vital to understand the velocity fluctuations in a turbulent flow. The turbulent shear stress formula is important for determining the friction velocity of the flow. We will discuss how to derive the turbulent shear stress equation from the Navier-Stokes equations in this article.

Turbulent Flow Layers and Shear Stress Variation 

The turbulence in pipe flow is common in engineering problems. For a better physical design of fluid flow systems, it is important to understand the velocity profile, friction factor, and pressure losses in a turbulent flow.

The velocity profile of fully developed turbulent flow is different from that of laminar flow. In the turbulent flow of constant viscosity-constant density fluid, the velocity gradient of the flow (as well as the wall shear stress) is greater than that of the laminar flow. The region close to the wall in a turbulent flow is dominated by viscous effects and showcases a linear velocity profile.

In the inner layer, known as the viscous wall layer or viscous sublayer, the velocity variation is linear with distance. The inner layer in a turbulent flow also satisfies the no-slip conduction at the wall. In the outer region or inertial sublayer of the turbulent flow, the velocity is nearly constant with distance from the wall. The velocity at the outer layer does not obey the no-slip condition at the wall. There is a third layer called the overlap layer, which is between the inner layer and outer layer.

The layer regions discussed above exhibit different shear stress, as shown in the table below.  

Turbulent flow layer and associated shear stress type

Deriving the Turbulent Shear Stress Equation From the Navier-Stokes Equations 

Shear stress is encountered by both laminar and turbulent flow. Due to the momentum transfer between randomly moving molecules in laminar flow, shear stress is experienced, and it is a microscopic phenomenon. However, in turbulent flow, shear stress is a macroscopic phenomenon resulting from the momentum transfer between finite-sized fluid particles that move randomly. The shear stress and physical properties of turbulent flow are different from laminar flow.

Let’s learn how to derive the turbulent shear stress equation and see how it is related to laminar shear stress.

Consider a turbulent flow with temporal and spatial variations in velocity components (u, v, w) and pressure (p) given by:

Velocity and pressure equation

The term —- gives the mean values and the term with ‘ gives the random fluctuations.

The equation for momentum in the Navier-Stokes equations for the u-component of momentum can be written as:

Equation for u-component of momentum

Using the continuity equation, equation (2) reduces to:

Simplified equation for u-component of momentum

Substituting equation (1) into equation (3), the momentum relation in the x-direction can be given as:

Equation showing momentum relation in the x-direction

Turbulent stress has the same dimensions as laminar stress and occurs simultaneously with laminar stress terms.

Turbulent Pipe Flow and Turbulent Shear Stress

In the pipe-boundary layer flow, shear stress is dominant, which is associated with the y-direction normal to the wall. The total shear stress can be obtained with the following equation: 

Total shear stress equation

Where:

Turbulent shear stress equation

In the wall layer of a turbulent pipe flow:

Turbulent shear stress relation with laminar shear stress in wall layer

In the outer layer of a turbulent pipe flow:

Turbulent  shear stress relation with laminar shear stress in the outer layer

In pipe flows, the turbulent shear stress equation is critical in determining the velocity distribution. Cadence can offer you CFD simulation tools that model fluid flow behavior in physical systems such as pipes and automobiles.

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