2 INTRODUCTION Model the laminar and turbulent flow over a flat plate Compare with the theoretical predictionsStudy of flow parameters like Nusselt Number, Skin Friction Coefficient, shear layer thickness, velocity and temperatureEffects of change in length and initial flow conditions
3 THEORY Boundary layer Assumptions: Steady, Incompressible, 2D, Laminar flowThe shear layer is thin. This is true if Re>>1In the boundary layer u i.e. the velocity in the x direction scales with Lv the velocity in the y direction scales with u>>v thereforeand to simplify the Navier Stokes equations to the corresponding boundary layer formsThe term for flow over a flat plate as constant.
4 THEORY Laminar - Blasius Similarity Solution Using the similarity variablewe haveSubstituing these variables into the x-momentum equation of the boundary layer we will obtain the following ODE as a function ofAssuming no slip conditions we have u(x,0)=v(x,0)=0 and the free stream merge condition u(x,)=UeThese convert to
5 THEORY Laminar - Blasius Equation Solution Parameters Exact from Blasius0.6641.72151.328
6 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 edgesAt very large x Re gets large and the flow gets separated.
9 Flolab Parameters Mesh Size Used: Medium Number Iterations: 1000 Fluids: Air and WaterFlat Plate Length: 1m and 5mRe # Range: 100 – 2.23*10^9
10 RESULTS – Velocity Profiles Laminar – AirShear stress and Cf decreases as x increasesbecomes thicker as x increasesAs 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 #
11 RESULTS – Velocity Profiles Turbulent – WaterThe difference with the laminar case is that the shear layer in turbulence is much thinner than its laminar counterpart.
12 RESULTS – Skin Friction (Cf) Laminar –Air/WaterWith the increase of x the value of Cf decreases as Rex increases. Cf 1/RexAs 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.
13 RESULTS – Skin Friction (Cf) Turbulent–Air/WaterWith 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.
14 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.
15 RESULTS – Reynolds Similarity Laminar –Air/WaterFor 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.
16 RESULTS – Skin Friction (Cf) Laminar – AirWith 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
17 RESULTS – Skin Friction (Cf) Turbulent – WaterWith x increasing the value ofso Cf*Ln^2(Re) becomes independent of xFlolab results showed a good agreement as far the constant, but the power of Re does not match Ln^2(Re)
18 RESULTS – Nusselt Number Laminar –WaterWith 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
19 RESULTS – Nusselt Number Laminar – Airfor 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 constantFlolab results showed a pretty good agreement as far the power of 0.5, but the constant does not match
20 RESULTS – Reynolds Similarity Laminar –Air/WaterNu 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
21 RESULTS – Nusselt Number Turbulent –Air/Waterfor turbulent case. Here for a constant Pr as Re increases with x, the nusselt number increases as well as Re4/5.
22 RESULTS – Nusselt Number Turbulent –Air/Waterfor turbulent case. Without heat transfer the empirical law is not valid. The flowlab results showed a steep increase of Nu, even when Tw = Te
23 RESULTS – Nusselt Number Turbulent –WaterWith x increasing the value ofso Nu/Re^0.8 becomes independent of x for a PrFlolab results showed a good agreement as far the constant, but the power of Re does not match Re^0.8
24 RESULTS – Temperature Profiles Laminar – WaterThe profiles closely matches the profile as found by PolhaussenThe temperature is almost constant when Tw=Te
25 RESULTS – Temperature Profiles Turbulent –Air/WaterThe thickness of the thermal boundary layer in the case of turbulence decreases as compared to the laminar case.
26 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.
27 RESULTS – Change of Length Change of Length from 1m to 5 mif 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.
28 RESULTS – Velocity Contour Sample velocity contour as generated by FLOLAB at different X locations
29 ConclusionsFlowlab shows fairly good agreement in all respects as compared to the theoretical values obtained by the boundary layer theoryHowever 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 .