 # Anoop Samant Yanyan Zhang Saptarshi Basu Andres Chaparro

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Anoop Samant Yanyan Zhang Saptarshi Basu Andres Chaparro
FLOLAB – FLAT PLATE Anoop Samant Yanyan Zhang Saptarshi Basu Andres Chaparro

INTRODUCTION Model the laminar and turbulent flow over a flat plate
Compare with the theoretical predictions Study of flow parameters like Nusselt Number, Skin Friction Coefficient, shear layer thickness, velocity and temperature Effects of change in length and initial flow conditions

THEORY Boundary layer Assumptions:
Steady, Incompressible, 2D, Laminar flow The shear layer is thin. This is true if Re>>1 In the boundary layer u i.e. the velocity in the x direction scales with L v the velocity in the y direction scales with  u>>v therefore and to simplify the Navier Stokes equations to the corresponding boundary layer forms The term for flow over a flat plate as constant.

THEORY Laminar - Blasius Similarity Solution
Using the similarity variable we have Substituing these variables into the x-momentum equation of the boundary layer we will obtain the following ODE as a function of Assuming no slip conditions we have u(x,0)=v(x,0)=0 and the free stream merge condition u(x,)=Ue These convert to

THEORY Laminar - Blasius Equation Solution Parameters
Exact from Blasius 0.664 1.721 5 1.328

THEORY Laminar - Limitations of B.L.
Near the edge the boundary layer theory fails as is not valid. At Re< 1000 Cf and non-dim pressure becomes high at leading and trailing edges At very large x Re gets large and the flow gets separated.

THEORY Definitions

THEORY Turbulence

Flolab Parameters Mesh Size Used: Medium Number Iterations: 1000
Fluids: Air and Water Flat Plate Length: 1m and 5m Re # Range: 100 – 2.23*10^9

RESULTS – Velocity Profiles
Laminar – Air Shear stress and Cf decreases as x increases becomes thicker as x increases As x increases we observe that the velocity exceeds the inlet velocity. In the case of the low Re # (10^2), is much thinner than a higher Re #

RESULTS – Velocity Profiles
Turbulent – Water The difference with the laminar case is that the shear layer in turbulence is much thinner than its laminar counterpart.

RESULTS – Skin Friction (Cf)
Laminar –Air/Water With the increase of x the value of Cf decreases as Rex increases. Cf  1/Rex As the value of inlet velocity is decreased so that the Re decreases then the skin friction coefficient goes up. For low Re i.e at 1500 or 100 the Cf does not follow the relation =0.664 as predicted by the Boundary layer solution by Blasius.

RESULTS – Skin Friction (Cf)
Turbulent–Air/Water With the increase of x the value of Cf decreases as Rex increases. Cf  As the value of inlet velocity is decreased so that the Re decreases the skin friction coefficient goes up.

RESULTS – Skin Friction (Cf)
Low Reynolds # With the increase of x the value of Cf decreases as Rex increases. Based on Dennis and Dunwoody Cf show a sharp increase at both the leading and trailing edge. Based on Flolab the leading edge does match but the trailing edge is drastically different. Both solutions are numerical solutions full Navier Stokes Eq.

RESULTS – Reynolds Similarity
Laminar –Air/Water For two different fluids water and air, varying viscosity and density, Re maintains the same value. At Re=10^5, Cf gives the same profile at different x as Cf depends on Re only. This shows that Re drives the problem in incompressible fluid flow.

RESULTS – Skin Friction (Cf)
Laminar – Air With x increasing the value of Cf*(Re)^0.5 becomes independent of x (theoretically) Flolab results showed a pretty good agreement as far the constant, but the power of Re does not match 0.5

RESULTS – Skin Friction (Cf)
Turbulent – Water With x increasing the value of so Cf*Ln^2(Re) becomes independent of x Flolab results showed a good agreement as far the constant, but the power of Re does not match Ln^2(Re)

RESULTS – Nusselt Number
Laminar –Water With x increasing the value of Nu/(Re)^0.5 becomes independent of x (theoretically for a particular Pr) With Twall=Te the profile doesn’t obey the Polhaussen Pr1/3 law

RESULTS – Nusselt Number
Laminar – Air for laminar case. Here for a constant Pr as Re increases with x,the nusselt number increases as well as Re, so Nu/Re should be a constant Flolab results showed a pretty good agreement as far the power of 0.5, but the constant does not match

RESULTS – Reynolds Similarity
Laminar –Air/Water Nu varies as Pr1/3.Pr of air is approx=0.72 and Pr for Water=6.So the Nuwater/Nuair=2.02.From the chart it is obvious that the flowlab results do agree with these for a constant Re=10^5

RESULTS – Nusselt Number
Turbulent –Air/Water for turbulent case. Here for a constant Pr as Re increases with x, the nusselt number increases as well as Re4/5.

RESULTS – Nusselt Number
Turbulent –Air/Water for turbulent case. Without heat transfer the empirical law is not valid. The flowlab results showed a steep increase of Nu, even when Tw = Te

RESULTS – Nusselt Number
Turbulent –Water With x increasing the value of so Nu/Re^0.8 becomes independent of x for a Pr Flolab results showed a good agreement as far the constant, but the power of Re does not match Re^0.8

RESULTS – Temperature Profiles
Laminar – Water The profiles closely matches the profile as found by Polhaussen The temperature is almost constant when Tw=Te

RESULTS – Temperature Profiles
Turbulent –Air/Water The thickness of the thermal boundary layer in the case of turbulence decreases as compared to the laminar case.

RESULTS – Transition Comparison of Laminar and Turbulent Model
Turbulent model predicts a much higher Nu and Cf than the laminar counterpart. We believe that at 106 the turbulent model must be the better one to solve the flow problem.

RESULTS – Change of Length
Change of Length from 1m to 5 m if the length of the plate and inlet velocity is changed such that Re is constant then Nu and Cf do not change at all as evident from the above graphs.

RESULTS – Velocity Contour
Sample velocity contour as generated by FLOLAB at different X locations

Conclusions Flowlab shows fairly good agreement in all respects as compared to the theoretical values obtained by the boundary layer theory However it is not possible to get *,  , directly from the flowlab results. The overshoot in the velocity profiles above the inlet velocity also couldn’t be explained .

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