Part 3 Flashcards

Question

What is the continuity equation in the Navier-Stokes equations?

Answer 1

∂𝑈/∂𝑥 + ∂𝑉/∂𝑦 + ∂𝑊/∂𝑧 = 0

Answer 2

To study viscous flows.

Answer 3

U∂U/∂x + V∂U/∂y + W∂U/∂z = -1/ρ ∂P/∂x + ν(∂²U/∂x² + ∂²U/∂y² + ∂²U/∂z²) + Fx/ρ

Answer 4

Viscosity is significant due to its effect on aircraft drag, particularly skin friction, which impacts fuel consumption and aircraft efficiency.

Answer 5

There is a direct link between drag and fuel consumption, making the understanding of viscous effects on drag crucial for efficient aircraft design.

Answer 6

Hydraulic systems, fuel systems, environmental systems.

Answer 7

Understanding viscous flows is important to identify the costs associated with fluid transport, leading to more efficient system designs.

Answer 8

Flow in pipes.

Answer 9

They often involve the transport of a fluid in pipes or tubes.

Answer 10

It introduces a cost associated with the transport or motion of fluids.

Answer 11

∂𝑈𝑖∂𝑥𝑖=0

Answer 12

𝑈𝑗∂𝑈𝑖∂𝑥𝑗=−1𝜌∂𝑃∂𝑥𝑖+∂∂𝑥𝑗(𝜈∂𝑈𝑖∂𝑥𝑗)+𝐹𝑖𝜌

Answer 13

A measure of a fluid's resistance to deformation and flow. ## Footnote Viscosity affects how internal stresses influence fluid motion.

Answer 14

* Laminar * Turbulent * Transitional

Answer 15

Motion in layers, where layers slide past each other without intermixing. ## Footnote Streamlines representing the flow do not cross paths.

Answer 16

Generally associated with low skin friction drag.

Answer 17

Tendency to separate from surfaces, leading to reduced lift and increased drag.

Answer 18

A complex flow regime characterized by chaotic and irregular fluid motion.

Answer 19

* 3D * Diffusive * Dissipative * Occurs at high Reynolds numbers * Many time and length scales * Chaotic

Answer 20

To characterize flow regimes and predict whether flow is laminar or turbulent.

Answer 21

An English professor known for his contributions to fluid mechanics, particularly the Reynolds number.

Answer 22

It provides a relative measure between inertial and viscous effects.

Answer 23

Approximately 2300.

Answer 24

The flow is likely to transition from laminar to turbulent.

Answer 25

It typically reduces.

Answer 26

𝑅𝑒=𝜌𝑈𝐷𝜇 or 𝑅𝑒=𝑈𝐷𝜈

Answer 27

Determining flow regime to select appropriate analysis tools.

Answer 28

By matching Reynolds numbers to reproduce the same flow physics in different geometries.

Answer 29

Flows that exhibit characteristics of both laminar and turbulent flow.

Answer 30

A line drawn by tracing the tangent of the velocity vector at any point in the flow field.

Answer 31

Fluid velocity and pipe diameter.

Answer 32

Extending laminar flow regions over aircraft to reduce drag.

Answer 33

The chaotic nature and many scales involved make it difficult to study and model.

Answer 34

To account for the effects of turbulence in aerospace engineering design.

Answer 35

Flow exhibits both laminar and turbulent characteristics.

Answer 36

Laminar and turbulent ## Footnote The Reynolds number is used to determine which regime is dominant.

Answer 37

Transitional flows ## Footnote Transitional flows are significant in aerospace applications.

Answer 38

To estimate the total drag ## Footnote Laminar flows generate significantly lower viscous drag compared to turbulent flows.

Answer 39

It influences aerodynamic behavior of aerofoils ## Footnote Particularly at low speeds (incompressible flows).

Answer 40

The effect of the Reynolds number on lift generated by aerofoils ## Footnote This includes the resulting lift coefficient.

Answer 41

𝐶𝑙=𝐿12𝜌𝑉2∞𝑆, 𝐶𝑑=𝐷12𝜌𝑉2∞𝑆 ## Footnote Where 𝐿 and 𝐷 are the lift and drag forces, 𝜌 is the fluid density, and 𝑉∞ is the freestream velocity.

Answer 42

Below approximately 4 degrees ## Footnote Above this angle, the lift coefficient increases with Reynolds number.

Answer 43

The flow is more turbulent ## Footnote Turbulent flow is beneficial to lift generation.

Answer 44

Positively ## Footnote Turbulence changes pressure distribution around the aerofoil, increasing lift.

Answer 45

Performance deteriorates ## Footnote This is counter-intuitive as laminar flow has lower viscous drag.

Answer 46

That lower viscous drag leads to higher 𝐿/𝐷 ## Footnote In reality, laminar flow can exhibit worse performance characteristics.

Answer 47

Explore thoroughly and without bias ## Footnote It is essential to check results against known flow physics.

Answer 48

It challenges intuition ## Footnote Results often contradict common expectations.

Answer 49

They affect performance characteristics of aerofoils ## Footnote Understanding boundary layers is crucial for analyzing flow features.

Answer 50

Viscosity results in fluid stresses which translate into a net loss of energy.

Answer 51

The velocity distribution inside a pipe is non-uniform due to viscosity.

Answer 52

The boundary layer is the region near a surface where viscous effects are significant, defined as the distance where velocity is 99% of the freestream value.

Answer 53

The boundary layer concept was proposed by Prandtl in 1904.

Answer 54

The no-slip condition states that the fluid velocity at the surface of a body is zero.

Answer 55

The boundary layer thickness, 𝛿, is defined as the distance from the surface where the velocity reaches 99% of the freestream velocity, 𝑈∞.

Answer 56

* Displacement thickness * Momentum thickness

Answer 57

Displacement thickness is the thickness by which the wall should be displaced for inviscid fluid to achieve the same mass flow per unit width.

Answer 58

𝑄′𝑣𝑖𝑠𝑐𝑜𝑢𝑠=∫∞0(𝑈∞−𝑢)𝑑𝑦.

Answer 59

Momentum thickness, 𝜃, is derived based on conservation of momentum and defined as 𝜃=∫∞0𝑢𝑈∞(1−𝑢𝑈∞)𝑑𝑦.

Answer 60

The Blasius boundary layer refers to laminar flow over a flat plate with zero-pressure gradient.

Answer 61

* Streamwise velocity component is larger than wall-normal component * Plate velocity in spanwise direction is zero * Pressure gradient in all directions is zero * Boundary layer thickness is much smaller than length of plate

Answer 62

* Continuity equation: ∂𝑢/∂𝑥 + ∂𝑣/∂𝑦 = 0 * Momentum equation: 𝑢∂𝑢/∂𝑥 + 𝑣∂𝑢/∂𝑦 = 𝜈∂²𝑢/∂𝑦²

Answer 63

The characteristic Reynolds number depends on the length of the plate, 𝐿.

Answer 64

𝛿(x)≈5x𝑅𝑒𝑥−−−√.

Answer 65

𝛿∗(x)≈1.721x𝑅𝑒𝑥−−−√.

Answer 66

𝜃(x)≈0.664x𝑅𝑒𝑥−−−√.

Answer 67

H=𝛿∗(x)/𝜃(x)≈1.721/0.664≈2.59.

Answer 68

𝜏𝑤≈0.332𝜇𝑈∞𝜈𝑥/𝑈∞−−−−−√.

Answer 69

The skin friction coefficient is defined as 𝑐𝑓=𝜏𝑤/(12𝜌𝑈²∞).

Answer 70

𝑐𝑓=0.664𝑅𝑒𝑥−−−√.

Answer 71

𝐷=∫𝐿0𝜏𝑤𝑑𝑎.

Answer 72

𝐷′=∫𝐿0𝜏𝑤𝑑𝑥.

Answer 73

𝐷′=𝜌𝑈²∞𝜃𝐿.

Answer 74

A small region of flow near solid boundaries where viscous effects dominate ## Footnote Characterized by large velocity gradients in the wall-normal direction

Answer 75

Exhibit large velocity gradients in the wall-normal direction

Answer 76

Velocity gradients are associated with fluid stresses which result in viscous drag

Answer 77

Laminar and turbulent boundary layers

Answer 78

The near-wall velocity gradient for the turbulent profile is much sharper than for the laminar boundary layer

Answer 79

𝜏𝑤𝑎𝑙𝑙=𝜇𝑑𝑢𝑑𝑦|𝑦=0

Answer 80

Far larger wall shear stresses

Answer 81

Considerably larger than for laminar boundary layers

Answer 82

A model for analyzing the turbulent boundary layer over a flat plate at zero-pressure gradient

Answer 83

𝑢∗=𝜏𝑤𝑎𝑙𝑙𝜌−−−−√

Answer 84

𝑢+=𝑢𝑢∗

Answer 85

Flow can be modeled using the relationship 𝑢+=𝑦+

Answer 86

𝑢+=1𝜅ln𝑦++𝐶+

Answer 87

𝜅=0.41, 𝐶+≈5

Answer 88

5<𝑦+<35

Answer 89

𝑢𝑈∞=(𝑦𝛿)17

Answer 90

𝛿∗=∫∞0(1−𝑢𝑈∞)𝑑𝑦

Answer 91

𝛿∗≈18𝛿

Answer 92

𝜃=∫∞0𝑢𝑈∞(1−𝑢𝑈∞)𝑑𝑦

Answer 93

𝜃≈772𝛿

Answer 94

𝜏𝑤𝑎𝑙𝑙=𝜌𝑈2∞𝑑𝜃𝑑𝑥

Answer 95

𝑐𝑓≈0.02𝑅𝑒−16𝛿

Answer 96

𝛿𝑥≈0.16𝑅𝑒1/7𝑥 or 𝑅𝑒𝛿≈0.16𝑅𝑒6/7𝑥

Answer 97

𝑐𝑓≈0.027𝑅𝑒1/7𝑥

Answer 98

𝐶𝐷=0.031𝑅𝑒1/7𝐿=76𝑐𝑓(𝐿)

Answer 99

The process by which a laminar boundary layer becomes turbulent

Answer 100

Freestream disturbances activating flow instabilities

Answer 101

Initial development waves in the natural transition route to turbulence

Answer 102

𝑅𝑒𝑡𝑟=𝑈∞𝑥𝑡𝑟/𝜈

Answer 103

𝑅𝑒𝑡𝑟

Answer 104

The distance measured from the leading edge where the transition process is expected to initiate.

Answer 105

Critical Reynolds number (𝑅𝑒𝑐𝑟𝑖𝑡)

Answer 106

5⋅10^5 < 𝑅𝑒𝑡𝑟 < 3⋅10^6

Answer 107

* Freestream turbulence intensity * Surface roughness

Answer 108

Because neither the laminar nor turbulent solutions should be used along the entire length of the plate.

Answer 109

Using laminar or turbulent solutions alone results in inaccurate estimations of viscous drag.

Answer 110

* Gas turbines * Aircraft aerodynamics * Wind turbines

Answer 111

The sensitivity of the transition process to external factors.

Answer 112

𝑇𝑢 = 𝑢′𝑟𝑚𝑠 / 𝑈∞ or 𝑇𝑢% = 100 * 𝑢′𝑟𝑚𝑠 / 𝑈∞

Answer 113

It moves upstream, decreasing the transitional Reynolds number.

Answer 114

* Gaps * Steps * Waviness

Answer 115

They promote transition to turbulence.

Answer 116

𝑘/𝛿∗, where 𝑘 is the step height and 𝛿∗ is the local displacement thickness.

Answer 117

The effect of freestream turbulence intensity.

Answer 118

Its influence on boundary layer transition and separation.

Answer 119

Lift producing devices such as wings cause induced drag due to downwash, leading to a shift in the lift vector that creates drag opposite to the flight direction. ## Footnote Induced drag is primarily caused by the pressure difference between the bottom and top of the wing.

Answer 120

Skin friction drag is caused by viscosity in the boundary layer, represented by wall shear stress. ## Footnote The equation Tw = M indicates the relationship with wall shear stress.

Answer 121

Pressure form drag is caused by pressure differences created by separated flow, leading to a pressure distribution component in the flight direction. ## Footnote This type of drag is dominant in bluff bodies such as spheres and cylinders.

Answer 122

Bluff bodies such as spheres and cylinders. ## Footnote The pressure form drag depends on the geometry of the aerofoil.

Answer 123

backwards shift in the lift vector

Answer 124

False ## Footnote Induced drag is also influenced by the pressure difference between the top and bottom surfaces of the wing.

Answer 125

Viscosity in the boundary layer.

Answer 126

It occurs when the pressure distribution has a component in the flight direction.

Answer 127

Skin friction drag ## Footnote This type of drag occurs due to the friction between the fluid and the surface of the object.

Answer 128

A shape that is not streamlined ## Footnote Examples include a sphere or a flat plate facing oncoming air.

Answer 129

Flow separates early due to blunt shape ## Footnote This leads to a large low-pressure wake forming behind the object.

Answer 130

Big pressure difference between front and back ## Footnote This results in a high drag coefficient (Cd value).

Answer 131

Flow accelerates and boundary layer stays attached ## Footnote This keeps the flow smooth and reduces drag.

Answer 132

Flow slows down and boundary layer likely separates ## Footnote This can lead to increased drag and turbulence.

Answer 133

It helps determine the characteristics of the boundary layer and flow separation ## Footnote Higher Reynolds numbers typically indicate turbulent flow.

Answer 134

Tw = 0 and viscosity constant for stationary flow ## Footnote These conditions help predict when separation will occur.

Answer 135

Turbulent or unsteady flow ## Footnote This leads to a larger wake and increased drag.

Answer 136

small pressure differences between top and bottom

Answer 137

Pressure or form drag ## Footnote This type of drag is affected by the shape's ability to maintain flow attachment.

Answer 138

Reattachment is possible ## Footnote This can help reduce drag in certain conditions.

Answer 139

It remains small ## Footnote This indicates efficient airflow around the object.

Answer 140

Laminar flow, flow is fully attached, very low drag ## Footnote This indicates that the flow remains smooth and orderly.

Answer 141

Re ~ 300 ## Footnote This marks the transition point where the flow starts to lose its attachment.

Answer 142

Burbulent boundary layer delays separation, lower drag ## Footnote This indicates that turbulent flows are generally more efficient in delaying flow separation.

Answer 143

Delays flow separation, lower chance of stalling ## Footnote High inertial forces help the flow stay attached to the surface.

Answer 144

Induce airflow to improve lift performance ## Footnote They help maintain attached flow and reduce the chance of stalling.

Answer 145

Stronger inertial forces ## Footnote This helps maintain attached flow over surfaces.

Answer 146

Cp drops sharply due to separation delay ## Footnote This indicates a significant change in drag characteristics.

Answer 147

Low Re drag is not influenced by velocity ## Footnote This means that drag remains consistent regardless of speed.

Answer 148

Transition to turbulence in boundary layer ## Footnote This is a critical point where flow characteristics change significantly.

Answer 149

Turbulent ## Footnote Turbulent flows tend to have different characteristics compared to laminar flows.

Answer 150

Viscous effects are high, flow separates earlier ## Footnote This indicates that the flow is more sensitive to surface shapes.

Answer 151

Low AOA ## Footnote AOA stands for Angle of Attack.

Answer 152

It is complex and cannot use steady state assumptions ## Footnote Vortex shedding refers to the periodic shedding of vortices from the surface of a body in a fluid flow.

Answer 153

They move hom experimentally ## Footnote 'Hom' likely refers to 'homo', indicating a uniform or consistent prediction.

Answer 154

The physics is complex ## Footnote This complexity can arise from various factors including fluid dynamics and turbulence.

Answer 155

* Vortex shedding * Wake fluctuations ## Footnote Wake fluctuations are variations in flow patterns that can affect the performance of a body in fluid dynamics.

Answer 156

Viscosity causes internal stresses and energy losses in fluid systems due to shear stress at surfaces. ## Footnote Understanding these losses is crucial for predicting fluid system performance.

Answer 157

Internal flows are fluid motions entirely bounded by surfaces, such as those found in pipes or ducts. ## Footnote Internal flows can be laminar or turbulent.

Answer 158

The no-slip condition states that the fluid velocity is zero at the walls of a channel due to viscosity. ## Footnote This leads to the formation of boundary layers at the surfaces.

Answer 159

Developing flow is characterized by a uniform velocity profile near the center and the formation of boundary layers at the walls. ## Footnote As the flow progresses, it transitions to fully developed flow.

Answer 160

Fully developed flow is when the velocity profile no longer changes as the flow moves downstream. ## Footnote This allows for simplifications in the governing equations.

Answer 161

∂𝑢/∂𝑥 + ∂𝑣/∂𝑦 + ∂𝑤/∂𝑧 = 0. ## Footnote This represents the conservation of mass.

Answer 162

ρ(u∂u/∂x + v∂u/∂y + w∂u/∂z) = -∂p/∂x + μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) + Fx. ## Footnote This represents the conservation of momentum.

Answer 163

Couette flow is the flow between two parallel plates, where one plate is moving and the other is stationary. ## Footnote It is used to analyze shear stress and viscosity.

Answer 164

* Incompressible and steady state * Infinite length and width of plates * 2-dimensional flow * Fully developed flow * Zero pressure gradient in the x-direction

Answer 165

μ∂²u/∂y² = 0. ## Footnote This leads to a linear velocity profile.

Answer 166

u(y) = (U/h)y. ## Footnote U is the velocity of the moving plate.

Answer 167

Planar Poiseuille flow is the flow between two stationary parallel plates driven by a pressure gradient. ## Footnote It contrasts with Couette flow where one plate is moving.

Answer 168

* Incompressible and steady state * Infinite length and width of plates * 2-dimensional flow * Fully developed flow * Constant pressure gradient in the x-direction

Answer 169

μ∂²u/∂y² = ∂p/∂x. ## Footnote This represents flow driven by a pressure gradient.

Answer 170

u(y) = (1/μ)(-dp/dx)(y²(h - y)). ## Footnote The maximum velocity occurs at y = h/2.

Answer 171

Hagen-Poiseuille flow is fully developed laminar flow in a circular pipe driven by a negative pressure gradient. ## Footnote It is analyzed using cylindrical coordinates.

Answer 172

* Incompressible and steady state * 2-dimensional flow * Fully developed flow * Constant pressure gradient in the x-direction

Answer 173

dp/dx = μr(d/dr)(r du/dr). ## Footnote This relates the pressure gradient to shear stress in the pipe.

Answer 174

u(r) = (1/4μ)(-dp/dx)(R² - r²). ## Footnote The maximum velocity occurs at r = 0.

Answer 175

u_avg = (1/2)u_max. ## Footnote This relates average velocity to maximum velocity.

Answer 176

Q = (πR^4/8μ)(-dp/dx). ## Footnote This shows how flow rate depends on pressure gradient and viscosity.

Answer 177

Measuring viscosity using concentric rotating cylinders. ## Footnote It is fundamental in rheometry.

Answer 178

Cooling systems and heat exchangers. ## Footnote It is relevant in thermal management.

Answer 179

Transport of fluids in pipes, such as refrigerants and fuel in aircraft. ## Footnote It is crucial for understanding fluid transport in engineering.

Answer 180

Turbulent internal flows, particularly in pipes. ## Footnote This will build on the understanding of laminar flows.

Answer 181

To develop a system to analyse, evaluate and appraise configurations involving flows in pipes and ducts ## Footnote This includes both laminar and turbulent flows.

Answer 182

Incompressible flows ## Footnote Compressible flows will be explored in a separate lecture.

Answer 183

Approximately 2000 ## Footnote Flow is considered laminar for Reynolds numbers below 2000 and turbulent for numbers above 4000.

Answer 184

𝑚˙=𝜌𝐴𝑉 ## Footnote Where 𝑚˙ is mass flow rate, 𝜌 is fluid density, 𝐴 is cross-sectional area, and 𝑉 is average velocity.

Answer 185

𝑄=𝐴𝑉 ## Footnote For a circular pipe, volumetric flow rate can also be expressed as 𝑄=𝜋𝑑²/4𝑉.

Answer 186

𝑅𝑒=𝜌𝑉𝑑𝜇 ## Footnote It can also be expressed as 𝑅𝑒=𝑉𝑑𝜈, where 𝜈 is the kinematic viscosity.

Answer 187

𝑑ℎ=4𝐴𝑃 ## Footnote Where 𝐴 is the cross-sectional area and 𝑃 is the wetted perimeter.

Answer 188

Head conservation in an ideal flow configuration ## Footnote It relates height, pressure, and velocity at two points in the flow.

Answer 189

* 𝐻𝑝 (energy addition from pumps) * 𝐻𝑓 (head losses due to friction) * 𝐻𝑚 (minor losses) ## Footnote These losses can significantly impact the overall energy balance in the system.

Answer 190

To determine the pressure loss in a pipe per unit length due to friction ## Footnote It is expressed as Δ𝑝𝐿=𝑓𝐷𝜌𝑉²/2𝑑.

Answer 191

𝑓𝐷=4𝑓 ## Footnote This means that if using the Fanning coefficient, the Darcy equations need to be adjusted by multiplying by 4.

Answer 192

Colebrook-White correlation ## Footnote It includes the effect of the pipe's internal surface roughness.

Answer 193

To simplify the analysis of pipe flow by presenting friction coefficients for various Reynolds numbers and roughness-to-diameter ratios ## Footnote It helps engineers quickly determine friction factors for both laminar and turbulent flow.

Answer 194

Losses through fittings, valves, elbows, etc. ## Footnote These losses are usually determined experimentally and can be significant.

Answer 195

ℎ𝑚=𝐾𝑉²/2𝑔 ## Footnote Where 𝐾 is the loss coefficient for the specific fitting or valve.

Answer 196

It determines whether the flow is laminar or turbulent, affecting the equations used for analysis ## Footnote Laminar flow equations apply for Re < 2000, and turbulent flow requires empirical correlations.

Answer 197

Increases the friction coefficient only when the flow is turbulent ## Footnote This effect is not significant in laminar flow.

Part 3 Flashcards

(232 cards)