TFL4801 Assignment 2 Thermo-Fluids 2026 Due Year 2026

UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology


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TFL4801: ThermoFlow

Assignment 2 — Boundary Layer Analysis and Flow Dynamics — 2026

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TFL4801
Module Code:
ThermoFlow
Module Name:
Boundary Layer Analysis and Flow Dynam-
Assignment Topic:
ics
Assignment 2
Assignment Number:
2026
Due Date:
50
Total Marks:




Submitted in partial fulfilment of the require-
ments for ThermoFlow (TFL4801) — UNISA 2026

,UNISA | TFL4801 Boundary Layer Analysis & Flow Dynamics



Question 1: Derivation of Displacement Thickness δ ∗ in a Boundary Layer

A boundary layer is formed due to flow of a viscous fluid on a flat plate. Derive
an expression for displacement thickness δ ∗ in the boundary layer that formed due
to viscous effects of the fluid.


1.1 Graphical Abstract: Boundary Layer on a Flat Plate


U∞
δ(x)U∞




u = 0 (no-slip)
δ∗

x1 Flat
x2 Plate x3


Figure 1: Boundary layer growth on a flat plate showing displacement thickness δ ∗ (Cengel
and Cimbala, 2014)



1.2 Physical Meaning of Displacement Thickness


When a viscous fluid flows over a flat plate, the no-slip condition at the wall causes the fluid
velocity to drop from the free-stream value U∞ to zero at the wall surface. The resulting ve-
locity deficit within the boundary layer causes a reduction in mass flow rate compared to an
inviscid flow occupying the same cross-section. The displacement thickness δ ∗ is defined as the
distance by which the external streamlines are displaced outward due to this velocity deficit
(White, 2011:467). Physically, if the boundary layer were replaced by a zero-velocity layer of
thickness δ ∗ , the mass flow rate would remain identical to the actual viscous case.


1.3 Derivation of Displacement Thickness δ ∗


Step 1: Establish the Mass Flow Rate Deficit


Consider a unit-width strip of the flow at a given cross-section at distance x from the leading
edge. For an inviscid (ideal) flow with no boundary layer, the mass flow rate per unit width
through a height δ is:




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,UNISA | TFL4801 Boundary Layer Analysis & Flow Dynamics



Z δ
ṁideal = ρ U∞ dy (1)
0


For the actual viscous flow, the mass flow rate per unit width is:


Z δ
ṁactual = ρ u dy (2)
0



Step 2: Define the Displacement Thickness


The displacement thickness δ ∗ represents the distance by which the inviscid streamlines are
displaced to account for the mass flow deficit. Equating the deficit to a fictitious inviscid stream
of thickness δ ∗ :


Z δ Z δ
ρ U∞ δ ∗ = ρ U∞ dy − ρ u dy (3)
0 0



Step 3: Simplify (Incompressible Flow, Constant Density)


For an incompressible flow (ρ = constant), dividing through by ρ U∞ :


Z δ  
∗ u
δ = 1− dy (4)
0 U∞

Since the velocity profile asymptotically approaches U∞ beyond δ, the upper limit may be ex-
tended to infinity without error (the integrand becomes zero for y > δ):


Z ∞ 
∗ u
δ = 1− dy (5)
0 U∞

This is the general expression for displacement thickness in a boundary layer formed by vis-
cous effects (Munson, Young and Okiishi, 2013:559).

Key Distinction
Displacement thickness vs. boundary layer thickness: The boundary layer thick-
ness δ is the total distance from the wall to where u ≈ 0.99 U∞ . The displacement
thickness δ ∗ is always less than δ and quantifies the effective outward shift of the exter-



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,UNISA | TFL4801 Boundary Layer Analysis & Flow Dynamics



nal flow caused by viscous retardation. For the Blasius laminar solution, δ ∗ ≈ δ/3.




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, UNISA | TFL4801 Boundary Layer Analysis & Flow Dynamics



Question 2: Derivation of Momentum Thickness θ in a Boundary Layer

For flow over a flat plate, derive an expression for momentum thickness “θ” in a
boundary layer that formed due to viscous effects of the fluid.


2.1 Graphical Abstract: Momentum Deficit in the Boundary Layer


U∞ Control Volume (unit width)
U∞


Inlet
Exit

θ

Flat Plate

Figure 2: Control volume for momentum deficit derivation of momentum thickness θ (White,
2011)



2.2 Physical Meaning of Momentum Thickness


The momentum thickness θ quantifies the reduction in momentum flux within the boundary
layer compared to an equivalent inviscid flow (Cengel and Cimbala, 2014:572). It is directly
related to the wall shear stress through the von Karman momentum integral equation, making
it a key parameter in boundary layer analysis and drag calculation.


2.3 Derivation of Momentum Thickness θ


Step 1: Momentum Flux in Inviscid Flow


For a unit-width section, the momentum flux in the free-stream (inviscid) flow through a strip
of height δ is:


Z δ
2
Ṁideal = ρ U∞ dy (6)
0



Step 2: Actual Momentum Flux in Viscous Flow


The actual momentum flux carried by the viscous boundary layer fluid is:

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