UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology
⋄
ASSIGNMENT 2
Boundary Layer Analysis and Flow Dynamics — 2026
⋄
Module Code: TFL4801
Module Name: ThermoFlow (Thermo-Fluids)
Assignment No.: Assignment 2
Due Date: July 2026
Semester: Year Module — 2026
Submitted in partial fulfilment of the requirements for ThermoFlow (TFL4801)
at the University of South Africa.
,UNISA | TFL4801 Thermo-Fluids Assignment 2
Question 1: Derivation of Displacement Thickness δ ∗ in a Boundary Layer
1.1 Graphical Abstract
y
U
U δ(x) — boundary
velocity layer edge
profile u(y)
U
U
U
displaced streamline
U
δ∗
x
Flat Plate
Figure 1: Boundary layer formation on a flat plate showing displacement thickness δ ∗
1.2 Physical Interpretation of Displacement Thickness
The displacement thickness δ ∗ quantifies the distance by which the external streamlines are
shifted outward due to the reduction in mass flow within the boundary layer (Munson, Young
and Okiishi, 2013). The viscous retardation of fluid near the wall reduces the mass flux com-
pared to an ideal (inviscid) flow. The solid surface is effectively displaced outward by δ ∗ to
account for this mass flow deficit (White, 2011).
1.3 Step-by-Step Derivation
Consider steady, incompressible, two-dimensional flow over a flat plate. Let U be the free-
stream velocity and u(y) be the actual velocity at height y within the boundary layer of thick-
ness δ.
Step 1: Mass flow rate in ideal (no boundary layer) flow.
For a strip of width W and height δ, the mass flow rate in an ideal flow is:
Z δ
ṁideal = ρU dy
0
Step 2: Actual mass flow rate within the boundary layer.
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,UNISA | TFL4801 Thermo-Fluids Assignment 2
Due to viscous retardation, the actual mass flow rate is:
Z δ
ṁactual = ρu dy
0
Step 3: Mass flow deficit.
The mass flow that is “missing” due to the boundary layer is:
Z δ
∆ṁ = ρ(U − u) dy
0
Step 4: Define displacement thickness.
The displacement thickness δ ∗ is defined as the thickness of an equivalent uniform-velocity
layer that carries the same mass flow deficit. Therefore:
Z δ
ρU δ ∗ = ρ(U − u) dy
0
Step 5: Final expression.
Dividing both sides by ρU (assuming incompressible flow, so density is constant):
Z δ
∗
u
δ = 1− dy
0 U
For a boundary layer that extends to infinity (since u → U as y → ∞, the integrand tends to
zero), the upper limit may be extended:
Z ∞
u
δ∗ = 1− dy
0 U
This expression is valid for any incompressible two-dimensional boundary layer, provided the
surface curvature is much larger than δ (Schlichting and Gersten, 2017).
Key Distinction
The displacement thickness δ ∗ does not represent the actual thickness of the boundary
layer. It is a measure of the outward displacement of the outer streamlines caused by
the mass flow deficit within the viscous region. For a laminar Blasius boundary layer
√
over a flat plate, δ ∗ ≈ 1.72x/ Rex .
Page 2 of 17
,UNISA | TFL4801 Thermo-Fluids Assignment 2
Question 2: Derivation of Momentum Thickness θ in a Boundary Layer
2.1 Graphical Abstract
y
U y
U δ(x)
U
Momentum deficit
θ x
Flat Plate
Figure 2: Momentum thickness θ as a measure of momentum deficit in the boundary layer
2.2 Physical Interpretation
The momentum thickness θ is the equivalent height of a fluid slab, moving at the free-stream
velocity U , that carries the same momentum deficit as the actual boundary layer (White, 2011).
It appears directly in the von Kármán momentum integral equation and is therefore central to
boundary layer analysis.
2.3 Step-by-Step Derivation
Step 1: Momentum flux in ideal flow.
Without a boundary layer, the momentum flux through a strip of height δ is:
Z δ
Ṗideal = ρU 2 dy
0
Step 2: Actual momentum flux.
The actual momentum flux within the boundary layer is:
Z δ
Ṗactual = ρu2 dy
0
Step 3: Momentum deficit.
However, the mass flow entering the boundary layer carries momentum at the free-stream ve-
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, UNISA | TFL4801 Thermo-Fluids Assignment 2
locity U . The momentum deficit is therefore not simply ρ(U 2 − u2 ) but must account for the
actual mass flux at each height:
Z δ
∆Ṗ = ρu(U − u) dy
0
This follows from: the incoming mass flux ρu dy carries ideal momentum ρu · U dy, but the
actual momentum is ρu2 dy; the deficit per unit strip is thus ρu(U − u) dy.
Step 4: Define momentum thickness.
The momentum thickness θ is defined so that a uniform stream of velocity U and height θ car-
ries the same momentum deficit:
Z δ
2
ρU θ = ρu(U − u) dy
0
Step 5: Final expression.
Dividing both sides by ρU 2 :
Z δ
u u
θ= 1− dy
0 U U
Extending to infinity (since the integrand vanishes as u → U ):
Z ∞
u u
θ= 1− dy
0 U U
The von Kármán momentum integral equation then relates θ to wall shear stress τw :
dθ τw
=
dx ρU 2
(Schlichting and Gersten, 2017). This equation is fundamental to the approximate integral
methods used to solve boundary layer problems without requiring the full Navier–Stokes solu-
tion.
Critical Consideration
θ is always smaller than δ ∗ for typical velocity profiles. For a Blasius laminar boundary
√ √
layer, θ ≈ 0.664x/ Rex , while δ ∗ ≈ 1.72x/ Rex , giving δ ∗ /θ ≈ 2.59, a ratio known as
Page 4 of 17
College of Science, Engineering and Technology
⋄
ASSIGNMENT 2
Boundary Layer Analysis and Flow Dynamics — 2026
⋄
Module Code: TFL4801
Module Name: ThermoFlow (Thermo-Fluids)
Assignment No.: Assignment 2
Due Date: July 2026
Semester: Year Module — 2026
Submitted in partial fulfilment of the requirements for ThermoFlow (TFL4801)
at the University of South Africa.
,UNISA | TFL4801 Thermo-Fluids Assignment 2
Question 1: Derivation of Displacement Thickness δ ∗ in a Boundary Layer
1.1 Graphical Abstract
y
U
U δ(x) — boundary
velocity layer edge
profile u(y)
U
U
U
displaced streamline
U
δ∗
x
Flat Plate
Figure 1: Boundary layer formation on a flat plate showing displacement thickness δ ∗
1.2 Physical Interpretation of Displacement Thickness
The displacement thickness δ ∗ quantifies the distance by which the external streamlines are
shifted outward due to the reduction in mass flow within the boundary layer (Munson, Young
and Okiishi, 2013). The viscous retardation of fluid near the wall reduces the mass flux com-
pared to an ideal (inviscid) flow. The solid surface is effectively displaced outward by δ ∗ to
account for this mass flow deficit (White, 2011).
1.3 Step-by-Step Derivation
Consider steady, incompressible, two-dimensional flow over a flat plate. Let U be the free-
stream velocity and u(y) be the actual velocity at height y within the boundary layer of thick-
ness δ.
Step 1: Mass flow rate in ideal (no boundary layer) flow.
For a strip of width W and height δ, the mass flow rate in an ideal flow is:
Z δ
ṁideal = ρU dy
0
Step 2: Actual mass flow rate within the boundary layer.
Page 1 of 17
,UNISA | TFL4801 Thermo-Fluids Assignment 2
Due to viscous retardation, the actual mass flow rate is:
Z δ
ṁactual = ρu dy
0
Step 3: Mass flow deficit.
The mass flow that is “missing” due to the boundary layer is:
Z δ
∆ṁ = ρ(U − u) dy
0
Step 4: Define displacement thickness.
The displacement thickness δ ∗ is defined as the thickness of an equivalent uniform-velocity
layer that carries the same mass flow deficit. Therefore:
Z δ
ρU δ ∗ = ρ(U − u) dy
0
Step 5: Final expression.
Dividing both sides by ρU (assuming incompressible flow, so density is constant):
Z δ
∗
u
δ = 1− dy
0 U
For a boundary layer that extends to infinity (since u → U as y → ∞, the integrand tends to
zero), the upper limit may be extended:
Z ∞
u
δ∗ = 1− dy
0 U
This expression is valid for any incompressible two-dimensional boundary layer, provided the
surface curvature is much larger than δ (Schlichting and Gersten, 2017).
Key Distinction
The displacement thickness δ ∗ does not represent the actual thickness of the boundary
layer. It is a measure of the outward displacement of the outer streamlines caused by
the mass flow deficit within the viscous region. For a laminar Blasius boundary layer
√
over a flat plate, δ ∗ ≈ 1.72x/ Rex .
Page 2 of 17
,UNISA | TFL4801 Thermo-Fluids Assignment 2
Question 2: Derivation of Momentum Thickness θ in a Boundary Layer
2.1 Graphical Abstract
y
U y
U δ(x)
U
Momentum deficit
θ x
Flat Plate
Figure 2: Momentum thickness θ as a measure of momentum deficit in the boundary layer
2.2 Physical Interpretation
The momentum thickness θ is the equivalent height of a fluid slab, moving at the free-stream
velocity U , that carries the same momentum deficit as the actual boundary layer (White, 2011).
It appears directly in the von Kármán momentum integral equation and is therefore central to
boundary layer analysis.
2.3 Step-by-Step Derivation
Step 1: Momentum flux in ideal flow.
Without a boundary layer, the momentum flux through a strip of height δ is:
Z δ
Ṗideal = ρU 2 dy
0
Step 2: Actual momentum flux.
The actual momentum flux within the boundary layer is:
Z δ
Ṗactual = ρu2 dy
0
Step 3: Momentum deficit.
However, the mass flow entering the boundary layer carries momentum at the free-stream ve-
Page 3 of 17
, UNISA | TFL4801 Thermo-Fluids Assignment 2
locity U . The momentum deficit is therefore not simply ρ(U 2 − u2 ) but must account for the
actual mass flux at each height:
Z δ
∆Ṗ = ρu(U − u) dy
0
This follows from: the incoming mass flux ρu dy carries ideal momentum ρu · U dy, but the
actual momentum is ρu2 dy; the deficit per unit strip is thus ρu(U − u) dy.
Step 4: Define momentum thickness.
The momentum thickness θ is defined so that a uniform stream of velocity U and height θ car-
ries the same momentum deficit:
Z δ
2
ρU θ = ρu(U − u) dy
0
Step 5: Final expression.
Dividing both sides by ρU 2 :
Z δ
u u
θ= 1− dy
0 U U
Extending to infinity (since the integrand vanishes as u → U ):
Z ∞
u u
θ= 1− dy
0 U U
The von Kármán momentum integral equation then relates θ to wall shear stress τw :
dθ τw
=
dx ρU 2
(Schlichting and Gersten, 2017). This equation is fundamental to the approximate integral
methods used to solve boundary layer problems without requiring the full Navier–Stokes solu-
tion.
Critical Consideration
θ is always smaller than δ ∗ for typical velocity profiles. For a Blasius laminar boundary
√ √
layer, θ ≈ 0.664x/ Rex , while δ ∗ ≈ 1.72x/ Rex , giving δ ∗ /θ ≈ 2.59, a ratio known as
Page 4 of 17
