# CFD Modelling of Boundary Layer

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## 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

### 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

• High mesh density in the normal direction of the wall is required for turbulent boundary layers as shown below
• To save computational cost  for resolution of  viscous sub-layer, salable wall functions are used during turbulent simulation

## CFD of Laminar Flow Over A Flat Plate

• For laminar flow over a flat plate, a two dimensional geometry 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 ReL=20,000, Pr=0.7, Tin=300 K, Tw=350 K
• U velocity profile

• V velocity profile

Momentum Boundary Layer

Thermal Boundary Layer

### Comparison of Boundary layer for Constant Temperature and Heat Flux

• CFD results comparec at ReL=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

### 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 ReD=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 dth 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

## 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

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