Boundary Layer

By Mohit Uniyal|Updated : June 15th, 2022

The boundary layer theory was first proposed by L.Prandtl in 1904. Boundary layer theory is used to determine the aerodynamic drag (FD) and aerodynamic lift by analyzing hydrodynamic boundary layer formation in flying vehicles. Boundary layer theory is also used to design the shape of the vehicle, for example after observing the shape and motion of the fish, the aerodynamic foil shape is given to Aeroplan and turbine blades because aerodynamic foil shape has less drag coefficient (CD). The value of the coefficient of drag can be determined by boundary layer theory. 

Boundary layer theory has great applications in heat transfer enhancement i.e., analysis of thermal boundary layer formation. The study of both hydrodynamic and thermal boundary layer formations over the solid surface gives the relationship between frictional resistance of fluid flow and heat transfer characteristics. In this article, get an overview of hydrodynamic and thermal boundary layer formation over the flat surface and internal flow through pipes.

Table of Content

What is a Boundary Layer in Boundary Layer Theory?

The boundary layer in boundary layer theory is a thin layer of fluid formed over the solid surface or internal flow through pipes due to velocity or temperature gradient. Boundary layer theory is also used to study the internal flow of fluids. The boundary layer is basically classified as 

  • Hydrodynamic Boundary Layer
  • Thermal Boundary Layer

Hydrodynamic Boundary Layer for External Flow

When the real fluid flows over the solid body, the fluid is adhered to the solid boundary and assumed no-slip condition. i.e., the velocity of the fluid particle which is adhered to the boundary becomes equal to the velocity of the boundary. If the boundary is stationary, the velocity of fluid particles which are adhered to the stationary boundary becomes zero. Boundary layer theory is to determine the loss of energy for fluid flow in channels. The velocity gradients will be developed perpendicular to the solid boundary. The velocity profile for the real fluid flow over the horizontal solid surface is shown below.

  • Hydrodynamic boundary layer thickness (δh) is defined as the distance which is measured perpendicular to the boundary up to which the velocity of fluid becomes 99% of free stream velocity (U). To get a hydrodynamic boundary layer first draw velocity profiles at every point on the solid surface and mark the point perpendicular to the boundary where the fluid has 0.99U. The locus of all these points is termed as Hydrodynamic boundary layer. The formation of the boundary layer over the horizontal solid surface/flat plate is shown in the figure below.

Hydrodynamic for Internal Flow

When fluid flows through the circular/non-circular pipe the velocity at the surface of the pipe is zero and the maximum velocity (umax) at the pipe centre. The average velocity (uavg) fluid is used to analyze the characteristics of fluid flow. The velocity profile for the fluid which is flowing through the pipe is shown below.

Boundary-Layer

Thermal Boundary Layer for External Flow

When the fluid with free stream temperature (T) is flowing over the hot solid surface with a temperature (Ts). The temperature gradients will be developed perpendicular to the solid boundary. The temperature profile for the real fluid flow over the horizontal solid surface is shown below.

  • Thermal boundary layer thickness (δt) is defined as the distance which is measured perpendicular to the boundary up to which the temperature of fluid becomes 99% of free stream temperature (T). To get the thermal boundary layer first draw temperature profiles at every point on the solid surface and mark the point perpendicular to the boundary where the fluid has a temperature of 0.99T. The locus of all these points is termed the thermal boundary layers. The formation of a thermal boundary layer over the horizontal solid surface/flat plate is shown in the figure below.

Thermal Boundary Layer for Internal Flow

When fluid flows through the hot circular/non-circular pipe the temperature at the surface of the pipe is the surface temperature of the hot pipe (Ts) and the minimum temperature (Tmin) at the pipe centre. The average temperature (Tavg) fluid is used to analyze the thermal characteristics of fluid flow. The temperature profile for the fluid which is flowing through the pipe is shown below.

The relationship between Hydrodynamic (δh) and Thermal (δt) Boundary Layer Thicknesses

L.Prandtl proposed the relationship between δand δt after analyzing different hydrodynamic and thermal boundary layer formations in different cases. Prandtl number is a dimensionless number and it is defined as the ratio of momentum diffusivity(ν) and thermal diffusivity(α). The relationship between hydrodynamic and thermal boundary layer thicknesses is given below.

δhPr1/3

  • Here, δ= Hydrodynamic boundary layer thickness
  • δt= Thermal boundary layer thickness
  • Pr= Prandtl number v/α

Results of Hydrodynamic and Thermal Boundary Layer for Laminar External Fluid Flow

For the laminar hydrodynamic boundary layer, first, the velocity profile is derived by using boundary conditions the boundary layer thickness (δh), wall and average shear stress, and local and average drag coefficient by using Von Karman and Blasius equations. When the fluid flows over the horizontal flat solid surface the results for laminar flow in the case of both hydrodynamic and thermal boundary layers are given in the table below.

Here,

δh= Hydrodynamic boundary layer thickness

δt= Thermal boundary layer thickness

u= Velocity of fluid flow

T= temperature of fluid flow.

τw= Wall shear stress

hx= Local heat transfer coefficient

Rex= Local Reynold’s number

Pr= Prandtl number

τavg= Average shear stress

hL= Average heat transfer coefficient

Cfx= Local skin friction coefficient 

Nux= Local Nusselt number

CD= Coefficient of drag

NuL= Average Nusselt number

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FAQs on Boundary Layer

  • Pressure forces will act in opposition to the direction of fluid flow due to an increase of fluid pressure in the direction of fluid flow. It creates the adverse pressure gradient (dP/dx>0) and results in the increase of boundary layer thickness in the direction of flow. The adverse pressure gradient (dP/dx>0)  causes the long stretch on the boundary layer so that it gets separated from the surface and moves into the main fluid stream. This phenomenon is called boundary layer separation.

  • The shape factor (H) of the boundary layer is defined as the ratio of displacement thickness * and momentum thickness . Shape factor (H) is used to determine the nature of the flow.

  • When fluid flows over the horizontal flat solid surface, Reynold’s number is used to differentiate laminar and turbulent boundary layer regions. If the critical Reynold’s number is less than 5×10^5 is defined as the laminar boundary layer region and flow will be in a streamlined/laminar manner in this region. If Reynold’s number is greater than 7×10^5 is defined as the turbulent boundary layer region and flow will be random in this region.

  • A fluid having more kinematic viscosity will form thicker HBL as compared to the fluid having lesser kinematic viscosity for a given free stream velocity. The expression for hydrodynamic boundary layer thickness(δ_h) in terms of the distance measured from the leading-edge (x), kinematic viscosity(θ) and free stream velocity U_ is given below.

    δ_h = 5x^(1/2)θ/√ (U_∞)

  • The main assumption in boundary layer theory is a no-slip condition, i.e., When the fluid particle adhered to the boundary there is no relative motion between the fluid particle and the boundary of the solid surface hence the velocity of fluid particles becomes equal to the velocity of the boundary. If the boundary is stationary, the velocity of fluid particles which are adhered to the boundary will be zero.

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