CFD Modelling of Boundary Layer

What are types of boundary Layer and how they are numerically modeled?

By 

Dr. Sharad Pachpute

 

Introduction to Boundary Layer

•  In fluid dynamic, boundary layer is an essential topic. Boundary layer is a geion around the body within it viscous forces are significant.
• A thin layer of fluid is  formed close to the solid surface where the gradient in velocity  or any scalar is significant. This thin region is called as boundary layer
• A boundary layer can be there due to gradients in velocity, temperature and concentration or species
• Depending on type of flow and geometry. The boundary layer is formed for external and internal flows

Types of boundary layer

• Laminar boundary layers can be loosely classified according to their structure and the circumstances under which they are created
• The thin shear layer which develops on an oscillating body is an example of a Stokes boundary layer,
• Blasius boundary layer:
  • For a 2D laminar flow over a flat plate
  • Boundary layer over a flat plate is simplified with a function of similarity variable only

• Boundary Layer equations are given as below
•  Falkner–Skan boundary layer
  • generalization of Blasius profile.
  • It is formed as a fluid has swirling motion and Coriolis effect (rather than convective inertia), balances viscous forces with  an Ekman layer forms

Laminar boundary layer

• • • It is formed near to leading edge of surface
    • Viscous forces are dominant

Turbulent boundary layer

• • • • It is formed at a certain distance from the leading edge of surface
      • Inertia forces are dominant

Layers of Turbulent Boundary Layer

• Turbulent boundary layer consists for three main layers formed in the direction normal to the wall: Viscous Sub-layer, Buffer Layer , Turbulent Region

• Friction velocity is calculated using the wall shear stress and fluid density

U^* = friction velocity = sqrt (wall shear stress/density) , m/s

• Non-dimensional distance and velocity are defined as :

Y⁺ = normal distance × U^*/kinematic viscosity

         U⁺ = local velocity / friction velocity

1. Non-dimensional velocity is plotted with non-dimensional distance
2. Three main layers of turbulent region are formed as shown below

 Viscous Sub-layer

• • viscous stress is dominant
  • The plot shows a linear variation :   U⁺ =Y⁺    
  • Requires very fine mesh to capture very steep gradient close to the wall

 Buffer Layer

• • both viscous and turbulent  stress exist
  • U⁺ = f(Y⁺)
  • Requires very fine mesh to capture very steep gradient near the wall

 Turbulent Region

• • • turbulent  stress is dominant
    • U⁺ = 1/k ln (Y⁺)+ B
    • Requires  course mesh to capture less gradient away from the wall

Resolution of turbulent Boundary Layer

• To save computational cost  for resolution of  viscous sub-layer, scalable wall functions are used during turbulent simulation

 

 

CFD of Laminar Flow Over A Flat Plate

• For laminar flow over a flat plate, a two dimenional gemetry considered for CFD simulation

Computational Domain 

• Inlet specified with a constant velocity for Reynolds number of 10000 based on the length of flat plate
• Outlet specified with a zero gradient
• Governing equations of mass, momentum (Navier Stokes eqn.) and energy for laminar flow are solved using finite volume method-based solver, ANSYS FLUENT.

 

 Governing Equation for 2D boundary Layer

Continuity:

Momentum equation:

Energy equation:

• The local heat transfer coefficient (HTC) can be expressed as:

 

Numerical Procedure in ANSYS FLUENT

• CFD simulations carried out using a commercial FVM solver  ANSYS FLUENT
• In all simulations, flow was considered to be steady laminar and two dimenional flow
• Pressure based  incompreesible  solver
• Boudnary condition:
  • Inlet velocity and fluid properties selected as per Reynols number for constant air properties ( Pr  =0.7)
  • For thermal boudnary, wall temperature and heat fluxes varied to study the effect of parameters
• Pressure Velocity Coupling  – SIMPLE Algorithm
• Discretization Scheme: QUICK
• Convergence Criterio: 10^-6

CFD Results for Laminar Boundary Layer Over a Flat Plate

Case study for constant Wall Temperature 

• Boundary conditions are set for Re_L=20,000, Pr=0.7, T_in=300 K, T_w=350 K
• U velocity profile

    

• V velocity profile

   

 

Momentum Boundary Layer

   

Thermal Boundary Layer

Thermal Boundary layer for Constant wall Temperature condition

 Thermal Boundary layer for Constant wall Heat Flux condition

Comparison of Boundary layer for Constant Temperature and Heat Flux

• CFD results comparec at Re_L=10,000 and Pr=0.7

 

Boundary Layer for Internal Flow

Geometry for axis Symmetric Flow

• When fluid flows thought a pipe, a region of velocity gradient is formed neat the wall.
• Axis-symmetric boundary is formed

• Geometry for Two-Dimensional Pipe Flow

 Governing Equations in Cylindrical Co-ordinates

• The governing equations for laminar flow in cylindrical co-ordinates system for axis-symmetric problem are given below, in two dimensional cylindrical co-ordinates,
• all derivatives with respect to circumferential directions are zero.

Where x is the axial coordinate, r is the radial co-ordinate; u and v are velocity components in z and r directions respectively.

Energy equation:

Velocity Boundary Layer for Internal Flow

a) Velocity Contours in laminar flow through the pipe at Re_D=100

(b) Velocity Vectors in laminar flow

 

C) Static Pressure

Convective Boundary Layer For Internal Flow

• a thermal boundary layer develops when surface temperature is different from fluid temperature
• The growth of d_th depends on whether the flow is laminar or turbulent

Case A: Laminar Internal Flow for Constant wall Heat Flux

• Local Nusselt Number Variotion along the length of pipe is shown below.
• Based on analytical solution, the Nusselt number is 4.36 for fully developed thermally flow. A same value is obtained usign numerical simulation for Laminar flow

 

 

•  Temperature Distribution at Re=100, Pr=0.7

• Velocity Vectors with T magnitude

• Temperature Profile

• Non-dimensional Temperature profiles

• Local Nusselt number variation at constant heat flux condition 

 

• After a certain distance from the inelt, the wall and mean tempetrature vary linaerly along the length of pipe
• The following figure shows reslsts for a constant heat flux condition

Case B: Laminar Internal Flow for Constant wall Temperature

• The wall is maintained at constant temperature ( Ts = 350K) for laminar flow at Re=100 and Pr=0.7
• Fluid temperature increases from inlet to outlet
• Thermal boundary Layer

• Temperature profiles

• Non-dimensional Temperature (NDT) profiles for constant wall temperature condition is given below
• After a certain non-dimensional distance from the inlet, the non-dimensional temperature remains constant

• Axial Variation of wall and mean temperature at constant temperature condition is shown below
• The mean temperature of fluid increases exponentially and the difference between wall and mean fluid temperature decreases from inlet to outlet

• Local Nusselt number variation for laminar internal flow at constant wall temperature condition is given below
• Based on analytical solution for lamiar internal flow at contant wall temperature , the Nusselt number is 3.66 for fully developed thermally flow. A similar value is obtained using the numerical simulation for Laminar flow

Comparison of the Non-dimensional Temperature (NDT) Profiles

• As the Nusselt number is a measure of NDT gradient at the wall
• More non-dimensional temperature (NDT) gradient is observed in the case of constant heat flux condition compared to that for constant temperature condition

 

You tube Video for CFD Analysis of Thermal Boundary Layer

• Watch the video on applications and CFD analysis of thermal boundary :  You tube Video for  CFD of Boundary Layer

Conclusion

• Based on CFD results, momentum and thermal boundary have been studied for constant wall and heat flux boundary conditions
• Boundary layer does not change significantly for constant values of  Reynolds number and Prandlts number
• CFD results compare well with Nusselt number values obtained analytically
• The Nusselt number ( non – dimensional temperature gradient)  is higher for constant heat flux compared to that for constant temperature wall condition