Analysis of Flat Plate Boundary Layer flows P

Analysis of Flat Plate Boundary Layer flows P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Understand Pure Wall Effects under in Perceivable Fluid Flows …. . …

FLAT PLATE BOUNDARY LAYER EQUATIONS SIMILARITY At all x > 0

Similarity Of Velcoity Profiles (x) is the local boundary layer thickness. Suppose the solution has the property that when u/U is plotted against y/ , a universal function is obtained, with no further dependence on x. Such a solution is called a similarity solution. Similarity is satisfied if a plot of u/U versus y/ defines exactly the same function regardless of the value of x. Similarity satisfied Similarity not satisfied 3

Discovery of SIMILARITY Variable For a solution obeying similarity in the velocity profile we must have where f 1 is a universal function, independent of x (position along the plate). Since we have reason to believe that We can rewrite any such similarity form as Note that is a dimensionless variable.

Approximate Analytical Method -1 • Polhausen Approximate Solution

Polhausen Boundary Layer Profile

Are these Equations Eligible for Similarity Solution Maybe, maybe not, you never know , Try to convert these equations into an eligible form. Introduce Stream function, (x, y). Note that the stream function satisfies continuity identically.

Discovery of Similarity Variable by The Method Of Guessing We want our stream function to give us a velocity u = / y satisfying the similarity variable Along with a similarity function for velocity profile So we could start-off by guessing where F is a similarity function. so that Will it work ? ? Lets see…. .

Is it a right Guess ? If we assume not OK OK then we obtain This form does not satisfy the condition that u/U is a function of alone. Similarity demands the first derivative F’( ) is to be a function of alone, but not F. But note the extra function in x via the term (U x)-1/2! So our first guess failed because of the term (U x)-1/2. Use this guess to generate a better guess.

Second Guess learnt from the first Guess This time we assume Now remembering that x and y are independent of each other and recalling the evaluation of / y, Thus the first guess helped in guessing the correct function this new form of that satisfies similarity in velocity! But this does not mean that we are done. We have to solve for the function F( ).

The Basic of Similarity Variable Our goal is to reduce the partial differential equation using and . To do this we will need the following basic derivatives:

Conversion of Third Order PD into OD The next steps involve tedious differential calculus, to evaluate the terms in the BL Stream Function equation. The third order Partial derivative is:

Conversion of Second Order PDs into ODs we now work out the two second order derivatives:

Summary of Conversion

ODE governing Flat Plate BL

BOUNDARY CONDITIONS The boundary conditions are But we already showed that

The Blasius Equation • The Blasius equation with the boundary conditions exhibits a boundary value problem. • However, the location of one boundary is unknown, though boundary condition is known. • However, using an iterative method, it can be converted into an initial value problem. • Assuming a certain initial value for F =0 , Blasius equation can be solved using Runge-Kutta or Predictor. Corrector methods

Runge’s Numerical Reults 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 2 F 0 0. 00166 0. 00664 0. 01494 0. 02656 0. 04149 0. 05974 0. 08128 0. 10611 0. 13421 0. 16557 0. 20016 0. 23795 0. 27891 0. 32298 0. 37014 0. 42032 0. 47347 0. 52952 0. 5884 0. 65003 F' 0 0. 03321 0. 06641 0. 0996 0. 13277 0. 16589 0. 19894 0. 23189 0. 26471 0. 29736 0. 32978 0. 36194 0. 39378 0. 42524 0. 45627 0. 48679 0. 51676 0. 54611 0. 57476 0. 60267 0. 62977 F'' 0. 33206 0. 33205 0. 33199 0. 33181 0. 33147 0. 33091 0. 33008 0. 32892 0. 32739 0. 32544 0. 32301 0. 32007 0. 31659 0. 31253 0. 30787 0. 30258 0. 29667 0. 29011 0. 28293 0. 27514 0. 26675

F F' F'' 2. 2 2. 3 2. 4 2. 5 2. 6 2. 7 2. 8 2. 9 3 3. 1 3. 2 3. 3 3. 4 3. 5 3. 6 3. 7 3. 8 3. 9 0. 7812 0. 85056 0. 9223 0. 99632 1. 07251 1. 15077 1. 23099 1. 31304 1. 39682 1. 48221 1. 56911 1. 65739 1. 74696 1. 83771 1. 92954 2. 02235 2. 11604 2. 21054 0. 68132 0. 70566 0. 72899 0. 75127 0. 77246 0. 79255 0. 81152 0. 82935 0. 84605 0. 86162 0. 87609 0. 88946 0. 90177 0. 91305 0. 92334 0. 93268 0. 94112 0. 94872 0. 24835 0. 23843 0. 22809 0. 21741 0. 20646 0. 19529 0. 18401 0. 17267 0. 16136 0. 15016 0. 13913 0. 12835 0. 11788 0. 10777 0. 09809 0. 08886 0. 08013 0. 07191

F F' F'' 4 4. 1 4. 2 4. 3 4. 4 4. 5 4. 6 4. 7 4. 8 4. 9 5 5. 1 5. 2 5. 3 5. 4 5. 5 2. 30576 2. 40162 2. 49806 2. 595 2. 69238 2. 79015 2. 88827 2. 98668 3. 08534 3. 18422 3. 2833 3. 38253 3. 48189 3. 58137 3. 68094 3. 7806 0. 95552 0. 96159 0. 96696 0. 97171 0. 97588 0. 97952 0. 98269 0. 98543 0. 98779 0. 98982 0. 99155 0. 99301 0. 99425 0. 99529 0. 99616 0. 99688 0. 06423 0. 0571 0. 05052 0. 04448 0. 03897 0. 03398 0. 02948 0. 02546 0. 02187 0. 01591 0. 01347 0. 01134 0. 00951 0. 00793 0. 00658

F F' F'' 5. 6 5. 7 5. 8 5. 9 6 6. 1 6. 2 6. 3 6. 4 6. 5 6. 6 6. 7 6. 8 6. 9 7 3. 88032 3. 98009 4. 07991 4. 17976 4. 27965 4. 37956 4. 47949 4. 57943 4. 67939 4. 77935 4. 87933 4. 97931 5. 07929 5. 17928 5. 27927 0. 99748 0. 99798 0. 99838 0. 99871 0. 99898 0. 99919 0. 99937 0. 99951 0. 99962 0. 99977 0. 99983 0. 99987 0. 99993 0. 00543 0. 00446 0. 00365 0. 00297 0. 0024 0. 00193 0. 00155 0. 00124 0. 00098 0. 00077 0. 00061 0. 00048 0. 00037 0. 00029 0. 00022

7. 1 7. 2 7. 3 7. 4 7. 5 7. 6 7. 7 7. 8 7. 9 8 8. 1 8. 2 8. 3 8. 4 8. 5 8. 6 8. 7 8. 8 F 5. 37927 5. 47926 5. 57926 5. 67926 5. 77925 5. 87925 5. 97925 6. 07925 6. 17925 6. 27925 6. 37925 6. 47925 6. 57925 6. 67925 6. 77925 6. 87925 6. 97925 7. 07925 F' 0. 99995 0. 99996 0. 99997 0. 99998 0. 99999 1 1 1 F'' 0. 00017 0. 00013 9. 8 E-05 7. 4 E-05 5. 5 E-05 4. 1 E-05 3. 1 E-05 2. 3 E-05 1. 7 E-05 1. 2 E-05 8. 9 E-06 6. 5 E-06 4. 7 E-06 3. 4 E-06 2. 4 E-06 1. 7 E-06 1. 2 E-06 8. 5 E-07

NOMINAL BOUNDARY LAYER THICKNESS Recall that the nominal boundary thickness is defined such that u = 0. 99 U when y = . By interpolating on the table, it is seen that u/U = F’ = 0. 99 when = 4. 91. Since u = 0. 99 U when = 4. 91 and = y[U/( x)]1/2, it follows that the relation for nominal boundary layer thickness is 4. 6 4. 7 4. 8 4. 9 5 5. 1 F 2. 88827 2. 98668 3. 08534 3. 18422 3. 2833 3. 38253 F' 0. 98269 0. 98543 0. 98779 0. 98982 0. 99155 0. 99301 F'' 0. 02948 0. 02546 0. 02187 0. 01591 0. 01347

DRAG FORCE ON THE FLAT PLATE Let the flat plate have length L and width b out of the page: b L The shear stress o (drag force per unit area) acting on one side of the plate is given as Since the flow is assumed to be uniform out of the page, the total drag force FD acting on the plate is given as The term u/ y = 2 / y 2 is given from as

The shear stress o(x) on the flat plate is then given as 0 0. 1 0. 2 0. 3 0. 4 0. 5 F 0 0. 00166 0. 00664 0. 01494 0. 02656 0. 04149 F' 0 0. 03321 0. 06641 0. 0996 0. 13277 0. 16589 F'' 0. 33206 0. 33205 0. 33199 0. 33181 0. 33147 0. 33091 But from the table F (0) = 0. 332, so that boundary shear stress is given as Thus the boundary shear stress varies as x-1/2. A sample case is illustrated on the next slide for the case U = 10 m/s, = 1 x 10 -6 m 2/s, L = 10 m and = 1000 kg/m 3 (water).

Variation of Local Shear Stress along the Length Sample distribution of shear stress o(x) on a flat plate: U = 0. 04 m/s L = 0. 1 m = 1. 5 x 10 -5 m 2/s = 1. 2 kg/m 3 (air) Note that o = at x = 0. Does this mean that the drag force FD is also infinite?

DRAG FORCE ON THE FLAT PLATE The drag force converges to a finite value!

The Flat Plate Boundary Layer & Experimental Exploration

Measurement of Wall shear Stress

Evolution of Flat Plate Boundary Layer Thickness

Similarity of An evolving FBL • The application of integral balances to the control volume shown in above figure delivers more similarity features. • the boundary layer displacement thickness • the boundary layer momentum deficiency thickness and • the energy deficiency thickness.

Impermeable Streamlines of A BL • The lower side should be the wall itself; hence the drag force will be exposed. • The upper side should be a streamline outside the shear layer, so that the viscous drag is zero along this line.

Integral View of A FPBL • Integral Method

Using continuity equation : The partial integration of the second term of the left side of above equation gives:

Total Momentum deficit across boundary layer

Physical Interpretation of

The Displacement Thickness • Conservation of mass is applied to this Engineering CV @SSSF: Assuming incompressible flow, this relation simplifies to

* is the engineering formal definition of the boundary-layer displacement Thickness and holds true for any incompressible flow.

Physical Interpretation of Momentum and displacement thicknesses The ratio of displacement thickness to momentum thickness is called the shape factor, is often used in boundary-layer analyses: