prof. dr.-ing. norbert ebeling...0. 1 boundary layer theory prof. dr.-ing. norbert ebeling practical...
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Boundary Layer Theory
Prof. Dr.-Ing. Norbert Ebeling
Practical Training
Determination of a laminar boundary layer thickness on the flat plate
M.Sc. Aleksandra Marcinek Prof. Dr. Norbert Ebeling Phone +49(0)2551 9 62719 Phone +49(0)2551 9 62446 Email: [email protected] Email: [email protected]
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1. Introduction and learning objectives ------------------------------- Fehler! Textmarke nicht definiert.
2. Hydrodynamic and thermal boundary layer ---------------------- Fehler! Textmarke nicht definiert.
3. Training procedures ------------------------------------------------------------------------------------------------ 6
3.1 Experimental setup 6
3.2 Experimental procedure 7
3.3 Safety issues 7
4. Evaluation of results & Report ---------------------------------------------------------------------------------- 8
5. Recommended literature ----------------------------------------------------------------------------------------- 10
6. Appendix -------------------------------------------------------------------------------------------------------------- 11
7.1 Research schematics 11
7.2 Basic statistical analysis of errors 12
Table of content
1. Introduction and learning objectives
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1. Introduction and learning objectives
When a viscous fluid flows along a fixed impermeable wall, an essential condition is that the velocity
at any point of the fixed surface is zero. The grade, to which this condition modifies the general
character of the flow, depends upon the value of the viscosity. The modifying effect appears to be
confined within regions adjacent to the solid surfaces - these areas are called boundary layers.
Within such layers, the fluid velocity changes rapidly from zero to its free-stream value, and this may
imply a steep gradient of shearing stress; as a consequence, not all the viscous terms in the equation
of motion will be negligible.
Boundary layer has a wide range of applications from household practices to aerospace, heat and
mass transfer, sports aerodynamics etc.
◼ Airplanes - design of airfoils are important in airplanes. The airfoils are designed as per the
need of boundary layer flow so that it can be advantageous as per the need of airplane. The
important thing is the angle of attack of airfoils. So the airfoils are designed in such a way
so that there is smooth take off and landing of the plane in normal conditions. While the
plane takes off it needs to overcome air resistance so the airfoil is designed accordingly so
that there is no separation of boundary layer but the same air resistance is used to ad-
vantage while landing because the plane needs to deaccelerate. Therefore the angle of
attack of airfoil changes which breaks the boundary layer and hence air drag is created
which helps in de-acceleration.
◼ Automobiles - an automobile with more streamlined body is more efficient than a less
streamlined body. A more streamlined body means the boundary layer of air flow will not
break from the body surface so less will be the form drag.
◼ Mass transfer of catalytic reactions.
◼ Heat transfer enhancement.
◼ Golf ball aerodynamics.
◼ Mixing enhancement.
◼ Species transport (e. g. blowing for cleaning dust).
The practical training is performed in order to:
◼ Connection of the theoretical knowledge and the practical approach of fluid mechanics re-
search.
◼ The experience of the boundary layer test rig operation.
◼ Transfer of the collected data into real information about the fluid mechanics system.
◼ Experimental determination of the thickness of the thermal laminar boundary layer on the
flat plate.
◼ Investigation of the shape of the boundary layer over the flat plate.
◼ Comparison of the empirical data with analytical solution according to Blasius model.
◼ Calculation of the shear stress and the skin-friction coefficient.
◼ Calculation of the local convective constant and the heat flux within the boundary layer.
2. Hydrodynamic and thermal boundary layer
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2. Hydrodynamic and thermal boundary layer
Since the momentum and the energy law of conservation are corresponding to each other, the Blasius
solution for laminar velocity profile in the boundary layer above a flat plate can be easily extended to
describe thermal boundary layer for a heat transfer.
Fig. 1. Velocity profiles along laminar boundary layer on the flat plate
Based on the continuity and Navier–Stokes equations, with introduced the assumptions about the
two-dimensional flow of incompressible fluid in steady state condition, the equation of motion within
the boundary layer can be simplified and introduced with boundary conditions in a dimensionless
form.
Blasius derived an exact solution to presented laminar boundary layer equations. It express the
thickness of the boundary layer in the function of the Reynolds number for a laminar flow. It points
the region of flow where the velocity is less than 99% of the free stream velocity.
Description of thermal boundary layer uses a similarly derived energy balance, where momentum
diffusivity (here kinematic viscosity) was replaced by thermal diffusivity.
2. Hydrodynamic and thermal boundary layer
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Fig. 2. Temperature profile along laminar boundary layer on the flat plate
Velocity and thermal boundary layer become equivalent, when the momentum and thermal diffusivity
are equal. It means that the Prandtl number, which is defined as the ratio of momentum diffusivity to
thermal diffusivity, equals unity. In this case a thickness of the thermal and the velocity boundary layer
is the same. Respectively, for Pr < 1 the thickness of boundary layer increases, and for Pr > 1, the
thickness of boundary layer decreases.
Fig. 3. Thermal boundary layer thickness
2. Hydrodynamic and thermal boundary layer
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For the fluids with Prandtl number bigger than 0.6, the thermal boundary layer thickness can be
approximated by following equation.
3. Training procedures
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3. Training procedures
3.1 Experimental setup
1 | Equipment
◼ Wind tunnel with the air distribution system
◼ Fan
◼ Positioning machine
◼ Heated, copper flat plate
◼ Thermoelement type N (d=0.25mm)
◼ Digital thermometer (accuracy of 0.1°C)
◼ Thermoanemometer
2 | The wind tunnel and its dimensions
Fig. 4. Scheme of the wind tunnel
3. Training procedures
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Figure 5: Measuring points on the plate
Fig. 6. Cartesian coordinates system for the flat plate
3.2 Experimental procedure
1 | Check the air distribution system in the end of the wind tunnel.
2 | Turn on the heating of the flat plate.
3 | Use personal protective equipment – ear protection!
4 | Turn on the fan and adjust demanded value of the free stream velocity.
5 | Introduce temperature sensor into the wind tunnel.
6 | Measure temperature in the function of the sensor position, T = f(y).
7 | Repeat (5), (6) and (7) for 4 different x-values.
3. Training procedures
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During the practical training, activities will be moderated by the tutor and some instructions will be
given orally.
3.3 Safety issues
◼ Please observe the safety regulations of the "Laboratory for Thermal and Mechanical
Process Engineering". These regulations depend on the laboratory.
◼ In each experiment, you use a different workstation. Always familiarize yourself with the
location of the safety equipment (such as fire extinguisher, eye shower, emergency shower,
emergency stop button, fire alarm, collection point, etc.) and the escape routes before each
test.
◼ Instructions from laboratory staff must be obeyed.
◼ Please keep your workplace clean and tidy.
◼ Do not put obstacles on escape routes with equipment, chairs or bags and jacks / coats.
◼ Pay attention to your fellow colleagues.
The experiment does not pose any extraordinary health hazards. Nevertheless, note the following
specific hazards:
◼ The pipelines are under pressure (report leaking leaks to the laboratory supervisor
immediately).
◼ Hot surfaces (use heat protection gloves if necessary).
4. Evaluation of results & Report
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4. Evaluation of results & Report
1 | Present experimental results with adequate measurement uncertainty.
2 | Calculate a thickness of the velocity boundary layer.
𝛿99(𝑣) =5 ∙ 𝑥
√𝑅𝑒𝑥
𝛿99(𝑣) = 5 ∙ √𝜐 ∙ 𝑥
𝑢∞
𝑅𝑒𝑥 =𝜌 ∙ 𝑢∞ ∙ 𝑥
𝜇=𝑢∞ ∙ 𝑥
𝜐
Where: ρ=f(T) – air density [kg/m3]
μ=f(T) – air dynamic viscosity [Pa∙s]
ν=f(T) – air kinematic viscosity [m2/s]
𝛿99(𝑇,𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑) =𝛿99(𝑣)
√𝑃𝑟3
Where: Pr=f(T) – Prandtl number for the air [-]
3 | Compare obtained values of thicknesses for the velocity and the thermal boundary layers,
determine the differences.
4 | Calculate a shear stress at the surface of the plate.
𝜏0,𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = (𝜕𝑢
𝜕𝑦)𝑦=0
= 0.332 ∗ 𝜂 ∗𝑢∞𝑥
∗ √𝑅𝑒𝑥
4. Evaluation of results & Report
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5 | Calculate a skin-friction coefficient.
𝑐𝑓,𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =𝜏0,𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦12𝜌𝑢∞
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6 | Calculate a local convective constant and the heat flux in the region of the boundary layer.
ℎ𝑥 = 0.332 ∙𝑘
𝑥∙ √𝑅𝑒𝑥 ∙ √𝑃𝑟
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𝑞
𝐴= −𝑘 (
𝜕𝑇
𝜕𝑦)𝑦=0
= ℎ𝑥 ∙ (𝑇𝑠 − 𝑇∞)
7 | List of symbols
◼ u – velocity component in the x direction [m/s]
◼ v – velocity component in the y direction [m/s]
◼ ν – kinematic viscosity of fluid [m2/s],
◼ δ – thickness of the boundary layer [m]
◼ Re – Reynolds number [-]
◼ Pr – Prandtl number [-]
◼ ρ – density of fluid [kg/m3]
◼ μ – dynamic viscosity of fluid [Pa·s]
◼ u∞ – free stream velocity [m/s]
◼ us – velocity on the surface [m/s]
◼ k – thermal conductivity [W/m·K]
◼ cp – specific heat capacity [J/kg·K]
◼ α – thermal diffusivity [m2/s]
◼ T – absolute temperature [K]
◼ T∞ – temperature of the free stream [K]
◼ Ts – temperature at the surface [K]
◼ τ0 – shear stress at the surface [Pa]
◼ cf – skin-friction coefficient [-]
◼ h – convective heat transfer coefficient [W/m2·K]
◼ q – heat flow [W]
◼ A – area of heat transfer [m2]
5. Recommended literature
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5. Recommended literature
1 | P.P. Puttkammer, Boundary Layer over a Flat Plate, B.Sc. Report, Faculty of Engineering
Technology, Engineering Fluid Dynamics at University of Twente, Enschede 2013.
2 | Kay Gemba, Measurement of Boundary Layer on a Flat Plate, California State University, Long
Beach 2007.
3 | H. Schlichting, K. Gersten, Boundary-Layer Theory; Fundamentals of Boundary–Layer Theory,
DOI 10.1007/978-3-662-52919-5_2.
4 | Glenn Research Center, National Aeronautics and Space Administration, Boundary Layer,
www.grc.nasa.gov.
5 | Fluid Mechanics, Prof. T. I. Eldho, Department of Civil Engineering, Indian Institute of
Technology, Bombay: Lecture - 31 Boundary Layer Theory and Applications.
6 | I-campus project, School-wide Program on Fluid Mechanics, Modules on High Reynolds Num-
ber Flows, K. P. Burr, T. R. Akylas & C. C. Mei, Chapter 2 - Two-dimensional laminar boundary
layers.
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6. Appendix
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6. Appendix
6.1 Research schematics
Tab. 1. Temperature measurement above the heated flat plate
Distance from the leading edge x =
Free-stream velocity u∞ =
y [mm] T [°C] T [°C] T [°C]
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6. Appendix
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6.2 Basic statistical analysis of errors
◼ Mean value
◼ Standard deviation
◼ Uncertainty in the mean value
◼ Measured value