1

CHAPTER 4

BOUNDARY LAYER FLOW:

APPLICATION TO EXTERNAL FLOW

4.1 Introduction

Navier-Stokes equations and the energy equation are simplified using the boundary layer concept.

  • Under special conditions certain terms in the equations can be neglected.
  • Two key questions:

(1)What are the conditions under which terms in the governing equations can be dropped?

(2)What terms can be dropped?

4.2 The Boundary Layer Concept: Simplification of the Governing Equations

4.2.1 Qualitative Description

  • Consider convection over a semi-infinite plate (Fig. 4.1).
  • Under certain conditions the effect of viscosity is confined to a thin region near the surface called the velocity or viscous boundary layer,.
  • Under certain conditions the effect of thermal interaction is confined to a thin region near the surface called the thermal boundary layer.
  • Conditions for the formation of the two boundary layers:
  • Velocity boundary layer conditions:

(1) Slender body

(2) High Reynolds number (Re > 100)

  • Thermal boundary layer conditions:

(1) Slender body

(2) High product of Reynolds and Prandtl numbers (Re Pr > 100)

(4.1)

NOTE:

  • Fluid velocity at the surface vanishes.
  • Large changes in velocity across
  • Large changes in temperature across
  • Viscosity plays no role outside

4.2.2 The Governing Equations

Assumptions: (1) Steady, (2) two-dimensional, (3) laminar, (4) uniform properties, (5) no dissipation, and (6) no gravity.

Governing equations:

(2.3)

(2.10x)

(2.10y)

(2.19)

4.2.3 Mathematical Simplification

  • Simplify the above equations based on boundary layer approximations.

4.2.4 Simplification of the Momentum Equations

(i) Intuitive Argument

  • Follow the intuitive argument leading to:

(4.2)

and

(4.3)

Therefore

pp(x)

It follows that

(4.4)

  • (2.10x) simplifies to the following boundary layer x-momentum equation

(4.5)

(ii) Scale Analysis

  • We start by assuming that

(4.6)

  • Follow scale analysis leading to equations (4.2)-(4.4)
  • Follow scale analysis leading to:

(4.14b)

  • (4.14b) shows that (4.6) is valid when
  • (4.14b) is generalized as

(4.16)

4.2.5Simplification of the Energy Equation

  • The energy equation for two-dimensional constant properties flow is

(2.19)

  • (2.19) is simplified for boundary layer flow using two arguments:

(i) Intuitive Argument

  • Follow the intuitive argument leading to:

(4.17)

  • (2.19) simplifies to the following boundary layer energy

(4.18)

(ii) Scale Analysis

  • We start by assuming that

(4.19)

  • Follow scale analysis leading to equations (4.17) and (4.18)
  • Follow scale analysis for the validity of (4.19). Two cases are considered:

Case (1):. Follow the argument leading to:

(4.24)

Thus

when (4.25)

  • The criterion for : Taking the ratio of (4.24) to (4.14b) gives

(4.27)

Thus

when (4.28)

Case (2):. Follow the argument leading to:

(4.31)

Thus

when (4.32)

  • The criterion for : Taking the ratio of (4.31) to (4.14b)

(4.33)

Thus

when (4.34)

4.3Summary of Boundary Layer Equations for Steady Laminar Flow

  • Review all assumptions leading to the following boundary layer equations:

Continuity:

(2.3)

x-Momentum:

(4.13)

Energy:

(4.18)

Note the following:

  • The continuity is not simplified.
  • Solution to inviscid flow outside the boundary layer gives pressure gradient needed in (4.13).
  • To include buoyancy effect, add [] to (4.13).

4.4 Solutions: External Flow

  • For constant properties, velocity distribution is independent of temperature.
  • First obtain the flow field solution and then use it to determine temperature distribution.
  • Laminar Boundary Layer Flow over Semi-infinite Flat Plate: Uniform Surface

Temperature

The basic problem is shown in Fig. 4.5.

  • The plate is at uniform temperature.
  • Upstream temperature is
  • Apply all the assumptions summarized in Section 4.3.
  • Governing equations: (continuity, momentum, and energy) are given in (2.3), (4.13), and (4.18).

(i) Velocity Distribution.

Determine:

  • Velocity and pressure distribution.
  • Boundary layer thickness.
  • Wall shear

(a) Governing equations and boundary conditions

(2.3)

(4.13)

The velocity boundary conditions are:

(4.35a)

(4.35b)

(4.35c)

(4.35d)

(b) Scale analysis: boundary layer thickness, wall shear and friction coefficient.

We showed that

(4.16)

Define Darcyfrictioncoefficient

(4.37a)

Follow scale analysis leading to

(4.37b)

(c)Blasius solution: similarity method

  • Equations (2.3) and (4.13) are solved analytically by Blasius.
  • For inviscid flow over flat plate

u = , v = 0, p == constant(4.38)

Thus the pressure gradient is

(4.39)

(4.39) into (4.13)

(4.40)

  • (2.3) and (4.40) are solved by the method of similarity transformation.
  • The basic approach is combining the two independent variables x and y into a single variable (x, y) and postulate that u/V depends on only.
  • For this problem the correct form of the transformation variable  is

(4.41)

  • Assume

(4.42)

  • Using (4.41) and (4.42), integration of the continuity gives

(4.43)

  • Transform all derivatives in terms of f and , substitute (4.42), (4.43)

(4.44)

  • Boundary conditions (4.35a-4.35d) transform to

(4.45a)

(4.45b)

(4.45c)

(4.45d)

NOTE:

Table 4.1
Blasius solution [1]
/ f / /
0.0 / 0.0 / 0.0 / 0.33206
0.4 / 0.02656 / 0.13277 / 0.33147
0.8 / 0.10611 / 0.26471 / 0.32739
1.2 / 0.23795 / 0.39378 / 0.31659
1.6 / 0.42032 / 0.51676 / 0.29667
2.0 / 0.65003 / 0.62977 / 0.26675
2.4 / 0.92230 / 0.72899 / 0.22809
2.8 / 1.23099 / 0.81152 / 0.18401
3.2 / 1.56911 / 0.87609 / 0.13913
3.6 / 1.92954 / 0.92333 / 0.09809
4.0 / 2.30576 / 0.95552 / 0.06424
4.4 / 2.69238 / 0.97587 / 0.03897
4.8 / 3.08534 / 0.98779 / 0.02187
5.0 / 3.28329 / 0.99155 / 0.01591
5.2 / 3.48189 / 0.99425 / 0.01134
5.4 / 3.68094 / 0.99616 / 0.00793
5.6 / 3.88031 / 0.99748 / 0.00543
6.0 / 4.27964 / 0.99898 / 0.00240
7.0 / 5.27926 / 0.99992 / 0.00022
8.0 / 6.27923 / 1.00000 / 0.00001
  • The momentum is transformed into an ordinary differential equation.
  • Boundary conditions (4.35c) and (4.35d) coalesce into a single condition.
  • (4.44) is solved by power series. The solution is presented in Table 4.1.
  • From Table 4.1 we obtain

(4.46)

Scaling gives

(4.16)

  • From Table 4.1 we obtain

(4.48)

Scaling gives

(4.37b)

(ii) Temperature Distribution.

Determine:

  • Temperature distribution.
  • Thermal boundary layer thickness.
  • Heat transfer coefficient h(x).
  • Nusselt number Nu(x).

(a)Governing equation and boundary conditions

(4.18)

The boundary conditions are:

(4.49a)

(4.49b)

(4.49c)

(b)Scale analysis: Thermal boundary layer thickness, heat transfer coefficient and Nusselt number

  • Return to the results of Section 4.2.5:

Case (1): (Pr <1)

(4.50)

Case (2): (Pr >1)

(4.51)

  • Scale analysis for h. Begin with

(1.10)

Using the scales, the above gives

(4.52)

Case (1): (Pr <1). Substituting (4.50) into (4.52)

, for Pr <1 (4.53)

Defining the local Nusselt numberas

(4.54)

Substituting (4.53) into (4.54)

, for Pr <1 (4.55)

Case (2): (Pr >1). Substituting (4.51) into (4.52)

, for Pr >1 (4.56)

The corresponding Nusselt number is

, for Pr >1 (4.57)

(c)Pohlhausen’s solution: Temperature distribution, thermal boundary layer thickness, heat transfer coefficient, and Nusselt number

  • Equation (4.18) is solvedanalytically by Pohlhausen using similarity transformation.

Define

(4.58)

(4.58) into (4.18)

(4.59)

Boundary conditions (4.49) become

(4.60a)

(4.60b)

(4.60c)

Combine x and y into a single variable (x, y) given by

(4.41)

Assume

Velocity components u and v in (4.59) are given by Blasius solution

(4.42)

(4.43)

(4.41)-(4.43) into (4.59)

(4.61)

Using (4.41), the three boundary conditions (4.60a-4.60c) transform to

(4.62a)

(4.62b)

(4.62c)

Integration details of (4.61) are found in Appendix B. The temperature solution is

(4.63)

  • Surface temperature gradient is

(4.64)

  • The integrals in (4.63) and (4.64) are evaluated numerically.
  • Boundary layer thickness is determined from Fig. 4.6. The edge of the thermal layer is defined as the distance y where . This corresponds to

, at (4.65)

The heat transfer coefficient h is determined using equation (1.10)

(1.10)

Using (4.41) and (4.58) into the above

(4.66)

Average heat transfer coefficient

(2.50)

Substituting (4.66) into (2.50) and integrating

(4.67)

The local Nusselt number is obtained by substituting (4.66) into (4.54)

(4.68)

The corresponding average Nusselt number is

(4.69)

Table 4.2 gives for various values of Pr.

, Pr < 0.05 (4.71a)

, 0.6 < Pr < 10 (4.71b)

, Pr >10 (4.71c)

4.4.2 Applications: Blasius Solution, Pohlhausen’s Solution, and Scaling

  • Review Examples 4.1, 4.2, and 4.3. They illustrate the application of Blasius solution, Pohlhausen’s solution, and scaling to the solution of convection problems.

4.4.3Laminar Boundary Layer Flow over Semi-infinite Flat Plate: Variable Surface Temperature

  • Surface temperature varies with axial distance x according to

(4.74)

  • Assumptions: see Section 4.3.

(i) Velocity Distribution. Blasius flow field solution is applicable to this case:

(4.42)

(4.43)

where the similarity variable is defined as

(4.41)

(ii) Governing Equations for Temperature Distribution. Based on the assumptions listed in Section 4.3, temperature is governed by energy equation (4.18)

(4.18)

The boundary conditions for this problem are:

(a)

(b)

(c)

(iii) Solution.Define as

(4.58)

Assume

(4.75)

Using (4.41)-(4.43), (4.58), (4.74) and (4.75), energy equation (4.18) transforms to (see Appendix C for details)

(4.76)

Boundary conditions (a)-(c) become

(4.77a)

(4.77b)

(4.77c)

Local heat transfer coefficient and Nusselt number are determined using (1.10)

(1.10)

Using (4.41), (4.58) and (4.72) into the above

Substituting into (1.10)

(4.78)

The average heat transfer coefficient for a plate of length L is defined in equation (2.50)

(2.50)

(4.78) into (2.50)

(4.79)

(4.78) into (4.54) gives the local Nusselt number

(4.80)

The corresponding average Nusselt number is

(4. 81)

  • Key factor: surface temperature gradient

(iv) Results. (4.76) is solved numerically subject to boundary conditions (4.77). Fig. 4.8 gives for three Prandtl numbers.

4.4.4 Laminar Boundary Layer Flow over a Wedge: Uniform Surface Temperature

  • Assumptions: listed in Section 4.3.
  • x-momentum equation:

(4.13)

  • Inviscid flow solution for is

(4.82)

C is a constant and is defined as

(4.83)

Application of (4.13) to the inviscid flow outside the viscous boundary layer, gives

Substituting into (4.13)

(4.84)

The boundary conditions are

(4.85a)

(4.85b)

(4.85c)

Solution to the velocity distribution is obtained by the method of similarity. Define as

(4.86)

Assume

(4.87)

Continuity equation (2.3), (4.86), and (4.87) give the vertical velocity component

(4.88)

(4.82) and (4.86)-(4.88) into (4.84)

(4.89)

This is the transformed momentum equation. Boundary conditions (4.85) transform to

(4.90a)

(4.90b)

(4.90c)

  • Solution. (4.89) is integrated numerically. The solution gives and These in turn give the velocity components and
  • Temperature distribution. Start with the energy

(4.59)

Boundary conditions

(4.60a)

(4.60b)

(4.60c)

Where is defined as

(4.58)

Assume

(4.75)

is defined in (4.86). (4.86)-(4.88) and (4.75) into (4.59) and (4.60)

(4.91)

(4.92a)

(4.92b)

(4.92c)

  • Solution. Separating variables in (4.91), integrating twice and applying boundary conditions (4.92), gives the temperature solution as

(4.93)

16

(4.93) gives the temperature gradient at the surface

(4.94)

  • The integrals in (4.93) and (4.94) are evaluated numerically.
  • Results for are given in Table 4.3

  • Heat transfer coefficient h and Nusselt number Equation (1.10) gives h

(1.10)

Using (4.58), (4.75) and (4.86) into (1.10) gives

(4.95)

(4.95) into (4.54) gives the Nusselt number

(4.96)

where is the local Reynolds number defined as

(4.97)