Analysis of non-lifting potential flow past a thin symmetric hydrofoil

NAME 400
PROJECT AND THESIS
Prepared By:
M. Mashrur Hussain (1312046)
Arafat Zaman(1312049)
Under the Supervision of
Dr. Md. Shahjada Tarafder
Professor,
Department of Naval Architecture & Marine
Engineering,
Bangladesh University of Engineering & Technology
(BUET).
Dhaka-1000, Bangladesh.

Title
Analysis of non-lifting potential flow past a thin
symmetric hydrofoil by finite difference method.

Potential Flow Characteristics
A potential flow must be inviscid, incompressible
and irrotational.
That implies, viscosity, 𝜇 = 0,
Density, 𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,
Curl of the velocity vector, 𝛻 × 𝑉 = 0

2-D Hydrofoil
NACA 0012 Hydrofoil

Objective
To solve non-lifting potential flow problem around
the two-dimensional hydrofoil by using finite
difference method.
To calculate the pressure distribution on the
surface of the 2-D hydrofoil.
To compare the numerical result with the result
available in literature.

Mathematical Modeling
Total Velocity :
V = V∞ + 𝛻∅
Where, 𝑉 is the total velocity, 𝑉∞ is the base flow and
∅ is the perturbation potential.
Governing equation:
𝛻2
∅ = 0

Boundary Condition at Inflow:

𝜕∅
𝜕x
= 0

𝜕∅
𝜕y
= 0

Boundary Condition at Outflow

𝜕∅
𝜕x
= 0

𝜕∅
𝜕y
= 0

Boundary Condition at Far Field

𝜕∅
𝜕x
= 0

𝜕∅
𝜕y
= 0

Boundary Condition at Body Surface
• At the body surface the total velocity is tangential.
So, there is no velocity component along the
normal to the body surface. So,
𝛁∅ 𝐓 ∙ n = Ui + 𝛻∅ ∙ n = 0

Simplifying Boundary Condition
on the Body Surface
• 𝛁∅ 𝐓 ∙ 𝑛 = 𝑈 𝑖 + 𝛻∅ ∙ 𝑛 = 0
𝑈𝑛 𝑥 +
𝜕∅
𝜕𝑥
𝑖 +
𝜕∅
𝜕𝑦
𝑗 𝑛 𝑥 𝑖 + 𝑛 𝑦 𝑗 = 0
𝑈𝑛 𝑥 +
𝜕∅
𝜕𝑥
𝑛 𝑥 +
𝜕∅
𝜕𝑦
𝑛 𝑦 = 0
𝑈𝑛 𝑥 +
𝜕∅
𝜕𝑥
𝑛 𝑥 + 𝑈
𝑑𝑌
𝑑𝑥
𝑛 𝑦 = 0 ; 𝑎𝑠,
𝝏∅
𝝏𝒚
= 𝑼
𝒅𝒀
𝒅𝒙
𝑓𝑜𝑟 0 < 𝑥𝑖 < 𝑐
𝑈 𝑛 𝑥 +
𝑑𝑌
𝑑𝑥
𝑛 𝑦 +
𝜕∅
𝜕𝑥
𝑛 𝑥 = 0
𝑆𝑜,
𝝏∅
𝝏𝒙
= −𝑼 𝟏 +
𝒅𝒀
𝒅𝒙
𝒏 𝒙
𝒏 𝒚
• For Thin Hydrofoil,
𝝏∅
𝝏𝒚
= 𝑼
𝒅𝒀
𝒅𝒙
𝑓𝑜𝑟 0 < 𝑥𝑖 < 𝑐 ; [ (Equation No: 9.32] of An Introduction to
Theoretical and Computational Aerodynamics By Jackie Moran

Discretized Equation at Inner Node
By applying the boundary condition and using the
Laplace equation discretized equation is developed.
Discretized equation for Inner Node:
ao∅o = aE∅E + aW∅W + aN∅N + aS∅S
aE =
1
hE
2
+ hEhW
; aW =
1
hW
2
+ hEhW
; aN =
1
hN
2
+ hNhS
; aS =
1
hS
2
+ hNhS
;
And, ao = aE + aW + aN + aS

Discretized Equations at Inflow
Discretized equations for inflow is divided into three
segments. That is,
Bottom corner
Side
Top corner
For Bottom corner, (i=1, j=1)
ao∅o = aE∅E + aN∅N
Where
aE =
1
hE
2
aN =
1
hN
2
ao =
1
hE
2 +
1
hN
2 = aE + aN

For Side (i = 1, j = 2, N − 1 )
ao∅o = aE∅E + aN∅N + aS∅S
Where,
aE =
1
hE
2
aN =
1
hN
2 +hS
2
aS =
1
hN
2 +hS
2
ao = aE + aN + aS

For Top Corner (i = 1, j = N)
ao∅o = aE∅E + aS∅S
Where,
aE =
1
hE
2
aS =
1
hS
2
ao = aE + aS

Discretized Equations at Far-field
For Node where i = 2, M − 1 , j = N
ao∅o = aS∅S + aE∅E + aW∅W
Where,
aS =
1
hS
2
aW = aE =
1
hE
2+hW
2
ao = aS + aW + aE

Discretized Equations at Outflow
Discretized equations for outflow is also divided into
three segments. That is,
Bottom corner
Side
Top corner
For Bottom corner (i = M, j = 1)
𝑎 𝑜∅ 𝑜 = 𝑎 𝑊∅ 𝑊 + 𝑎 𝑁∅ 𝑁
Where,
𝑎 𝑊 =
1
ℎ 𝑊
2
𝑎 𝑁 =
1
ℎ 𝑁
2
𝑎 𝑜 =
1
ℎ 𝑊
2 +
1
ℎ 𝑁
2 = 𝑎 𝑊 + 𝑎 𝑁

For Side (i = M, j = [2, N − 1])
𝑎 𝑜∅ 𝑜 = 𝑎 𝑊∅ 𝑊 + 𝑎 𝑁∅ 𝑁 + 𝑎 𝑆∅ 𝑆
Where,
𝑎 𝑊 =
1
ℎ 𝑊
2
𝑎 𝑁 =
1
ℎ 𝑁
2 +ℎ 𝑆
2
𝑎 𝑆 =
1
ℎ 𝑁
2 +ℎ 𝑆
2
𝑎 𝑜 = 𝑎 𝑊 + 𝑎 𝑁 + 𝑎 𝑆

For Top Corner(i = M, j = N)
𝑎 𝑜∅ 𝑜 = 𝑎 𝑊∅ 𝑊 + 𝑎 𝑆∅ 𝑆
Where,
𝑎 𝑊 =
1
ℎ 𝑊
2
𝑎 𝑆 =
1
ℎ 𝑆
2
𝑎 𝑜 =
1
ℎ 𝑊
2 +
1
ℎ 𝑆
2 = 𝑎 𝑊 + 𝑎 𝑆

Discretized Equation for Symmetry line
Without Body Surface
ao∅o = aN∅N + aE∅E + aW∅W
Where,
aN =
1
hN
2
aE =
1
hE hE+hW
aW =
1
hW hE+hW
ao = aN + aE + aW

Discretized Equation on the Body
Surface
∅o
2
hE
2
+ hW
2 +
1
hN
2 =
∅E
hE
2
+ hW
2 +
∅W
hE
2
+ hW
2 +
∅N
hN
2 +
𝐔 𝟏 +
𝐝𝐘
𝐝𝐱
𝐧 𝐱
𝐧 𝐲
𝐡 𝐄 − 𝐡 𝐖
𝐡 𝐄
𝟐
+ 𝐡 𝐖
𝟐 −
𝐡 𝐍 𝐔
𝐝𝐘
𝐝𝐱
𝐡 𝐍
𝟐
ao∅o = aE∅E + aW∅W + aN∅N + S
Where,
aE =
1
hE
2+hW
2 and S =
𝐔 𝟏+
𝐝𝐘
𝐝𝐱
𝐧 𝐱
𝐧 𝐲
𝐡 𝐄−𝐡 𝐖
𝐡 𝐄
𝟐+𝐡 𝐖
𝟐 −
𝐡 𝐍 𝐔
𝐝𝐘
𝐝𝐱
𝐡 𝐍
𝟐
aW =
1
hE
2+hW
2
aN =
1
hN
2
ao = aE + aW + aN

Grid Generation
Structured Algebraic H type grid is used. The
advantages are:
 Easy to implement
 Good efficiency
 With a smooth grid transformation, discretization
formulas can be written on the same form as in the
case of rectangular grids.

Installing Body Surface in the Grid
The formula for the shape of a NACA 00xx foil, with
“xx” being replaced by the percentage of thickness
to chord is:
Source:https://en.wikipedia.org/wiki/NACA_airfoil
yt = 5t 0.2969 x − 0.1260x − 0.3516x2 + 0.2843x3 − 0.1036x4
Where,
x is the position along the chord from 0 to 1.00
yt is the half thickness
t is the maximum thickness

Pressure Coefficient
• Matrix A and Matrix B were Created.
• By using matrices solver ∅ values were evaluated.
• Pressure co-efficient was determined by using
the following formula, CP = −
2
V∞
𝜕Φ
𝜕𝑥 𝑦=0

Cp vs (X/C)
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1 1.2
PRESSURECOEFFICIENT,CP
NON DIMENSIONAL X CO-ORDINATE
X/C
PRESSURE DISTRIBUTION COMPARISON ON THE THIN-HYDROFOIL
Gregory & O'Reilly Experiment Value Numerical Result
NACA 0012 HYDROFOIL
ANGLE OF ATTACK = 0 DEGREE

Discussion
• “The thin airfoil approximation costs you accuracy,
especially near stagnation points. However, when
the geometry is as complicated as it is for real
aircraft, body-conforming meshes are hard to
construct that approximations are sometimes
necessary for the finite-difference method to be
usable. This is the main advantage of the panel
method, its ability to describe accurately flows past
realistic geometries with relative ease.” – Jack
Moran, An Introduction to Theoretical and
Computational Aerodynamics, Page No: 303

Analysis of non-lifting potential flow past a thin symmetric hydrofoil

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Analysis of non-lifting potential flow past a thin symmetric hydrofoil

Similar to Analysis of non-lifting potential flow past a thin symmetric hydrofoil (20)

Recently uploaded

Recently uploaded (20)

Analysis of non-lifting potential flow past a thin symmetric hydrofoil