The Boundary Layer Equation in Inviscid Flow

Key Takeaways

• Inviscid flow has zero viscous force due to which the boundary layer formed is so thin that the pressure near the boundary and outside the boundary is the same.  

• The Euler equation can be used as the boundary layer equation in inviscid flow, given all the boundary conditions are specified (such as the no-slip condition). 

• The Euler equation for inviscid flow helps predict flow behavior and the onset of turbulence, which is beneficial for making complex design optimizations. 

Inviscid fluid flow around an airfoil

Viscosity is a crucial fluid property that influences fluid behavior and boundary layer formation. Viscosity causes the velocity of the flowing fluid to slow down as it comes in contact with a solid surface and experiences a frictional force. The velocity decreases from free stream to zero near the surface, where a thin layer is formed, called the boundary layer. 

But what happens when the fluid does not have any viscosity? In inviscid flow, lack of viscosity means the boundary layer formed is so thin that it can be considered non-existent, i.e., the pressure near the surface and beyond it is the same. But the solid surface still influences the flow.  In this article, we will take a look at the boundary layer equation in inviscid flow to explore how boundary conditions affect fluid behavior and how CFD can help analyze this behavior. 

Inviscid Flow and Boundary Conditions

Inviscid flow refers to the type of fluid flow where viscous forces can be considered negligible, i.e., the friction between the fluid and the surface in contact is zero. Thus, there is no shearing stress in such flow and only normal stress can be considered during the analysis. Such flow models can be used in theoretical analysis of the flow behavior in fluid applications, including in aerodynamic design, weather pattern prediction, or hydrodynamic analysis. 

Given the lack of viscosity, the boundary layer equation for inviscid flow is inapplicable. In such a case, the flow field can be analyzed using the Euler equation, given the boundary conditions are properly specified. The Euler equation is based on the no-slip boundary condition for inviscid flow, which indicates the velocity of the fluid at the boundary is zero. 

The general boundary layer equation can be expressed using the Navier-Stokes equation: 

Here, υ is the kinematic viscosity, ρ is the fluid density, and P is the pressure of the fluid. u₁ and u₂ are the velocity along the directions x₁ and x₂, respectively. 

For inviscid flow, the above equation can be reduced to:

U is the velocity of the fluid.

The above Euler equation facilitates an understanding of velocity and pressure distribution near the boundary when the flow is inviscid. The velocity is low close to the surface and continuously increases upstream until it reaches the free stream velocity. The study of this flow velocity and the associated Reynolds number helps in the prediction of flow patterns and the onset of turbulence, which is beneficial in aerodynamic and hydrodynamic design. 

Predicting Boundary Layer Behavior With CFD Analysis of Inviscid Flow

CFD has been a popular tool among system designers to predict the flow behavior of complex system designs. While inviscid flow may not be practical for a real-world application, the simplified flow model can help to make initial estimates of the fluid behavior so the areas of optimization can be identified. Here’s how CFD helps analyze boundary layer equations in inviscid flow:

Use CFD Simulations to Solve the Boundary Layer Equation in Inviscid Flow 

Numerical analysis of boundary layer parameters in inviscid flow can be done by solving the set of partial differential equations associated with the flow. CFD solvers can help in this analysis by modeling and analyzing the inviscid flow using Euler’s equation. The results obtained from the simulation are key in determining the velocity and pressure distribution in the fluid system, which is important to understanding key changes in the flow; for instance, separation, turbulence, shock waves, and vortices. Using CFD for numerical simulation and analysis, engineers can easily solve theboundary layer equation in inviscid flow, facilitating the evaluation and optimization of complex fluid systems. 

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