Characterization of Unsteady Boundary Layer Response to Free-Stream VariationsKanika Gakhar
2.29 Final Project
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT2
Motivation: Effect of Combustor Turbulence on Boundary Layer Dissipation for High-Pressure Turbine Blades
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT3
Project Objectives
• Model Free Stream
Turbulence (FST) as
a periodic time and
space varying free-
stream velocity Define variables to
isolate and model
effect of various
FST parameters on
BL response
• Characterize
unsteady response
of laminar
boundary layer
over a semi-infinite
flat plate
Laminar
BL
Flat Plate
Rate of Convection
Amplitude
Wavenumber & Frequency
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT4
Project Scope & Overview
Navier-Stokes ⇒ Thin Shear Layer Eqns.
Finite-Difference Discretization Scheme
+ Newton Iteration Solver
Steady Free-
Stream
Variations
Unsteady
Free-Stream
Variations
Integral Boundary
Layer Method
(closure relations)
Analytical solution
2.29 Finite-Volume
Code
Quantify effect of free-stream variations
on laminar boundary layer dissipation
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT5
Boundary Layers: Brief Background
Most aerodynamic flows have high Reynolds numbers
Solution of Navier Stokes equations exhibits boundary layers at solid-surface boundaries
magnitude of 𝑉(𝑟) rapidly drops from the bulk-flow velocity down to 𝑉 = 0 at the surface
Origin of boundary layer behavior highest-derivative term 𝛻2𝑉 being multiplied by small
viscosity coefficient, 𝜈
Image Credits: Mark Drela, Aerodynamics of Viscous Fluids
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT6
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑢
𝜕𝑢
𝜕𝑥=𝜕𝑝
𝜕𝑥+ 𝜇
𝜕2𝑢
𝜕𝑦2+𝜕2𝑢
𝜕𝑥2
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT7
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑢
𝜕𝑢
𝜕𝑥=𝜕𝑝
𝜕𝑥+ 𝜇
𝜕2𝑢
𝜕𝑦2+𝜕2𝑢
𝜕𝑥2
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑒𝑢𝑒𝜕𝑢𝑒𝜕𝑥
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT8
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑒𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT9
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
Step 2: Write 𝑢 and 𝑣 in terms of stream-function, 𝜓, to impose continuity
𝜕 𝜌𝑢
𝜕𝑥+𝜕 𝜌𝑣
𝜕𝑦= 0
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT10
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
Step 2: Write 𝑢 and 𝑣 in terms of stream-function, 𝜓, to impose continuity
𝜕 𝜌𝑢
𝜕𝑥+𝜕 𝜌𝑣
𝜕𝑦= 0 ⇒ 𝜌𝑢 =
𝜕𝜓
𝜕𝑦, 𝜌𝑣 = −
𝜕𝜓
𝜕𝑥
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT11
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
Step 2: Write 𝑢 and 𝑣 in terms of stream-function, 𝜓, to impose continuity
𝜕 𝜌𝑢
𝜕𝑥+𝜕 𝜌𝑣
𝜕𝑦= 0
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑒𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
⇒ 𝜌𝑢 =𝜕𝜓
𝜕𝑦, 𝜌𝑣 = −
𝜕𝜓
𝜕𝑥
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT12
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
Step 2: Write 𝑢 and 𝑣 in terms of stream-function, 𝜓, to impose continuity
𝜕 𝜌𝑢
𝜕𝑥+𝜕 𝜌𝑣
𝜕𝑦= 0
𝜕
𝜕𝑡
𝜕𝜓
𝜕𝑦+𝜕𝜓
𝜕𝑦
𝜕𝑢
𝜕𝑥−𝜕𝜓
𝜕𝑥
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑒𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
⇒ 𝜌𝑢 =𝜕𝜓
𝜕𝑦, 𝜌𝑣 = −
𝜕𝜓
𝜕𝑥
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT13
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
Step 2: Write 𝑢 and 𝑣 in terms of stream-function, 𝜓, to impose continuity
𝜕 𝜌𝑢
𝜕𝑥+𝜕 𝜌𝑣
𝜕𝑦= 0
𝜕
𝜕𝑡
𝜕𝜓
𝜕𝑦+𝜕𝜓
𝜕𝑦
𝜕𝑢
𝜕𝑥−𝜕𝜓
𝜕𝑥
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑒𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕𝜏
𝜕𝑦
⇒ 𝜌𝑢 =𝜕𝜓
𝜕𝑦, 𝜌𝑣 = −
𝜕𝜓
𝜕𝑥
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT14
Thin-Shear Layer Equation
Step 1: Write pressure term in terms of inviscid boundary-layer edge velocity
𝜕 𝜌𝑢
𝜕𝑡+ 𝜌𝑢
𝜕𝑢
𝜕𝑥+ 𝜌𝑣
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕
𝜕𝑦𝜇𝜕𝑢
𝜕𝑦
Step 2: Write 𝑢 and 𝑣 in terms of stream-function, 𝜓, to impose continuity
𝜕 𝜌𝑢
𝜕𝑥+𝜕 𝜌𝑣
𝜕𝑦= 0
𝜕
𝜕𝑡
𝜕𝜓
𝜕𝑦+𝜕𝜓
𝜕𝑦
𝜕𝑢
𝜕𝑥−𝜕𝜓
𝜕𝑥
𝜕𝑢
𝜕𝑦=
𝜌𝑒𝜕𝑢𝑒𝜕𝑡
+ 𝜌𝑒𝑢𝑒𝜕𝑢𝑒𝜕𝑥
+𝜕𝜏
𝜕𝑦
BC at edge: 𝑢 = 𝑢𝑒BC on solid body: 𝑢 = 0, 𝜓 = 0
⇒ 𝜌𝑢 =𝜕𝜓
𝜕𝑦, 𝜌𝑣 = −
𝜕𝜓
𝜕𝑥
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT15
Local Scaling Transformation: Independent Variable Transformation
Non-dimensionalize 𝑥 w.r.t. arbitrary plate length, 𝐿and 𝑦 w.r.t. boundary layer thickness scale,
𝛿 𝑥, 𝑡 =𝜈𝑥
𝑢𝑒 𝑥,𝑡∶
𝜉 =𝑥
𝐿𝜂 =
𝑦
𝛿 𝑥, 𝑡
−1
𝛿
𝜕𝜓
𝜕𝜂
𝜕𝛿
𝜕𝑡+
𝜕
𝜕𝑡
𝜕𝜓
𝜕𝜂+1
𝐿
𝜕𝜓
𝜕𝜂
𝜕𝑢
𝜕𝜉−1
𝐿
𝜕𝜓
𝜕𝜉
𝜕𝑢
𝜕𝜂
=𝜌𝑒𝑢𝑒𝛿
𝐿
𝜕𝑢𝑒𝜕𝜉
+ 𝜌𝑒𝛿𝜕𝑢𝑒𝜕𝑡
+𝜕𝜏
𝜕𝜂 Image Credits: Mark Drela, Aerodynamics of Viscous Fluids
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT16
Local Scaling Transformation: Dependent Variable Transformation
Non-dimensionalize 𝑢 w.r.t. edge velocity, 𝑢𝑒; 𝜓w.r.t. mass-flow scale, 𝑚; 𝜏 w.r.t. edge dynamic
pressure ∶
𝐹 =𝜓
𝑚𝑈 =
𝑢
𝑢𝑒𝑆 =
𝜏
𝜌𝑢𝑒2
𝜉𝐿
𝛿
𝑚 = 𝜌𝑒𝑢𝑒𝛿
𝜉𝐿
𝑢𝑒
𝜕𝑈
𝜕𝑡+𝑈 − 1
𝑢𝑒
𝜕𝑢𝑒𝜕𝑡
+𝜕𝐹
𝜕𝜂
𝜕𝑢
𝜕𝜉−𝜕𝐹
𝜕𝜉
𝜕𝑈
𝜕𝜂
= 𝛽𝑚𝐹𝜕𝑈
𝜕𝜂+ 𝛽𝑢 1 − 𝑈
𝜕𝐹
𝜕𝜂+𝜕𝑆
𝜕𝜂Image Credits: Mark Drela, Aerodynamics of Viscous Fluids
𝛽𝑢 =𝜉
𝑢𝑒
𝑑𝑢𝑒𝑑𝜉
𝛽𝑚=𝑑𝑙𝑛 𝑚
𝑑𝑙𝑛 𝜉
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT17
Local Scaling Transformation Equations
Three BCs:
BC at edge: 𝑈(𝜂 = 1) = 1
BC at body: 𝑈(𝜂 = 0) = 0
𝐹(𝜂 = 0) = 0
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT18
Discretized Equations: Finite-Difference Scheme
• t: Fully implicit, 1st order Backward Euler
• 𝜂: 1st order Forward FD, Trapezoidal Integral
• 𝜉: 1st order Backward FD, Trapezoidal Integral
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT19
Discretized Equations: Finite-Difference Scheme
• t: Fully implicit, 1st order Backward Euler
• 𝜂: 1st order Forward FD, Trapezoidal Integral
• 𝜉: 1st order Backward FD, Trapezoidal Integral
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT20
Discretized Equations: Finite-Difference Scheme
space
march
• t: Fully implicit, 1st order Backward Euler
• 𝜂: 1st order Forward FD, Trapezoidal Integral
• 𝜉: 1st order Backward FD, Trapezoidal Integral
time
march𝒕
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT21
Solving Non-Linear Residual Equations: Newton Iteration with Underrelaxation
• Sparse, banded matrix can reduce
operation count from 𝑂[𝑁3] to 𝑂[𝑁] by using
special sparse-matrix methods, such as
banded or block-tridiagonal solvers that
exploit zeros
• Underrelaxation prevent possible
divergence: 𝜔 ≤ 1
𝐹𝑗𝑛+1 = 𝐹𝑗
𝑛 + 𝜔 𝛿𝐹𝑗𝑈𝑗𝑛+1 = 𝑈𝑗
𝑛 + 𝜔 𝛿𝐹𝑗𝑆𝑗𝑛+1 = 𝑆𝑗
𝑛 +𝜔 𝛿𝐹𝑗
𝛽𝑢𝑗𝑛+1 = 𝛽𝑢𝑗
𝑛 + 𝜔 𝛿𝐹𝑗
• Convergence max𝑗
𝛿𝐹𝑗 , 𝛿𝑈𝑗 , 𝛿𝑆𝑗 < 𝜖
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT22
Results: Validation for Steady Case
Test case#1:
• 2D steady, laminar, flat-plate boundary
layer
• Favorable pressure gradient:
𝐶𝑝 =𝑥
𝐿
2
− 2𝑥
𝐿• 𝑅𝑒𝐿 = 106
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT23
Results: Validation for Steady Case with Spatial Variations
Test case#1:
• 2D steady, laminar, flat-plate boundary
layer
• Favorable pressure gradient:
𝐶𝑝 =𝑥
𝐿
2
− 2𝑥
𝐿• 𝑅𝑒𝐿 = 106
• Added variations in free-stream velocity
with amplitude, 𝐴 = 0.05, such that:
𝑢𝑒𝑢0
= 1 − 𝐶𝑝 + 𝐴 ∗ sin 2𝜋𝑥
𝐿
𝐿
𝜆
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT24
Results: Validation for Steady Case with Spatial Variations
Test case#1:
Compared solutions obtained from:
1. Finite-difference Boundary Layer
Code
2. Integral Boundary Layer Method
using Drela’s closure relations
3. Schlichting’s Analytical solution
o 𝐶𝑑 as a function of 𝑅𝑒𝜃 and
Pohlausen pressure gradient
parameter, 𝜆
• Solutions for Finite-Difference & Integral
Boundary Layer Codes in Agreement with
Analytical Soln.
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT25
Results: Validation for Uniform Unsteady Case
Test case#2:
• 2D unsteady, laminar, flat-plate boundary layer
• Uniform pressure distribution with no spatial variation
• 𝑅𝑒𝐿 = 106
• Oscillating inlet conditions, such that:𝑢
𝑢0ቚ𝑖𝑛𝑙𝑒𝑡
= 1 + 𝐴 ∗ cos 𝜔𝑡
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT26
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT27
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT28
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT29
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT30
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT31
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT32
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT33
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT34
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT35
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT36
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT37
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT38
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT39
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT40
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT41
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT42
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT43
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT44
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT45
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT46
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT47
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT48
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT49
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT50
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT51
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT52
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT53
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT54
Results: Validation for Uniform Unsteady Case
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT55
Results: Validation for Unsteady Case with Spatial Variations
Test case#3:
• 2D unsteady, laminar, flat-plate boundary layer
• 𝑅𝑒𝐿 = 106
• Traveling wave conditions, such that:𝑢𝑒𝑢0
= 1 + 𝐴 ∗ cos 𝜔𝑡 − 𝑘𝑥
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT56
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT57
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT58
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT59
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT60
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT61
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT62
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT63
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT64
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT65
Results: Validation for Unsteady Case with Spatial Variations – Finite Difference
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT66
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT67
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT68
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT69
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT70
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT71
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT72
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT73
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT74
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT75
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT76
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT77
Results: Validation for Unsteady Case with Spatial Variations – Finite Volume
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT78
Conclusion: Explaining Discrepancies & Potential Issues
• Finite-difference code doesn’t have an interactive boundary-layer component if pressure
gradient is not favorable, detects numerical flow separation and doesn’t produce physical
results code needs more work
• Rate of unsteady response of BL to fluctuating BCs does not match up between finite-
difference and finite volume comparative analysis needs further investigation
• Velocity profiles from two codes agree for certain time stamps and disagree for others
analytical & numerical assessment needs further work to understand discrepancies
• BCs imposed in two different codes don’t always physically represent the same BCs test
cases need more exploration for BCs
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT79
Conclusion: Advantages of Finite-Difference Method
Integral Boundary Layer
Methods
2.29 Finite Volume Code Thin-Shear Layer Finite
Difference Method
Relies on closure relations & curve
fit functions results not valid for
large free-stream variations & fully
unsteady response
Makes no assumptions and doesn’t
rely on closure relations (laminar)
results valid for any range of
space- and time- variations in free-
stream velocity
Only makes thin-shear layer
assumption results valid for any
range of space- and time- variations
in free-stream velocity
Requires only one grid point in y-
direction solves for integrated BL
characteristics
Requires grid for entire domain
solves for entire domain
Scales grid based on BL thickness
only solves for flow inside BL
Flow separation issues if
interacting BL algorithm not
implemented
Uses integrated values + closure
relations computational cost &
time extremely low
Uses uniform grid computational
cost increases as mesh needs to be
refined near wall
Uses scale transformed grid
computational cost & time
significantly lower
- CFL condition triggered need
extremely small time step to match
small mesh size
Fully-implicit scheme stable even
for large time steps
2.29: NUMERICAL FLUID MECHANICS
FINAL PROJECT80
Conclusion: Future Work
• Debug issues and assess finite difference results as compared to finite volume results
• Introduce vertical disturbances that are more characteristic of free-stream turbulence
• Translate analysis to turbulent boundary layers
• Translate analysis to high-pressure turbine blade geometry
• Incorporate combustor turbulence data from LES models as input condition for
unsteady BL code
Laminar BL
No-Slip Wall
Rankine VortexSlip Wall
StrengthSpatial Distribution
Rate of Convection
Thank You!Prof. Lermusiaux and Wael for all the help with my project and for answering my incessant questions
2.29: NUMERICAL FLUID MECHANICS FINAL PROJECT 81