9.3 Laminar Flat-Plate Boundary Layer: Exact Solution w-19

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1 9.3 Laminar Flat-Plate Boundary Layer: Exact Solution w-19 Laminar Flat-Plate Boundary Layer: Exact Solution The solution for the laminar boundary layer on a horizontal flat late was obtained by Prtl s student H. Blasius [2] in 198. For two-dimensional, stea, incomressible flow with zero ressure gradient, the goerning equations of motion (Eqs..27) reduce to [3] ð9:3þ ð9:4þ with boundary conditions at y ; u ; at y N; u ð9:þ Equations , with boundary conditions Eq. 9. are a set of nonlinear, couled, artial differential equations for the unknown elocity field u. To sole them, Blasius reasoned that the elocity rofile, u/, should be similar for all alues of x when lotted ersus a nondimensional distance from the wall; the boundary-layer thickness,, was a natural choice for nondimensionalizing the distance from the wall. Thus the solution is of the form u g ðηþ where η ~ y Based on the solution of Stokes [4], Blasius reasoned that ~ η y νx We now introduce the stream function, ψ, @x ffiffiffiffiffiffiffiffiffiffiffiffi νx= set ð9:6þ ð9:7þ ð:4þ satisfies the continuity equation (Eq. 9.3) identically. Substituting for u into Eq. 9.4 reduces the equation to one in which ψ is the single deendent ariable. Defining a dimensionless stream function as f ðηþ ffiffiffiffiffiffiffiffiffi ψ νx ð9:8þ makes f(η) the deendent ariable η the indeendent ariable in Eq With ψ defined by Eq. 9.8 η defined by Eq. 9.7, we can ealuate each of the terms in Eq The elocity comonents are gien ffiffiffiffiffiffiffiffiffi df νx dη νx df dη ð9:9þ ffiffi " ν f 2 ffiffiffiffiffiffiffiffiffi df νx 2 x 1 2 η ffiffi ν f x 2 x

2 w-2 Chater 9 External Incomressible Viscous Flow or 1 ffiffi ν 2 x f By differentiating the elocity comonents, it also can be shown that 2 2x η d2 ffiffiffiffiffiffiffiffiffiffiffiffi d 2 f d 3 f νx dη 3 Substituting these exressions into Eq. 9.4, we obtain with boundary conditions: 2 d3 f dη 3 1 f d2 f ð9:11þ at η ; at η-n; f df dη df dη 1 ð9:12þ The second-order artial differential equations goerning the growth of the laminar boundary layer on a flat late (Eqs ) hae been transformed to a nonlinear, third-order ordinary differential equation (Eq. 9.11) with boundary conditions gien by Eq It is not ossible to sole Eq in closed form; Blasius soled it using a ower series exansion about η matched to an asymtotic exansion for η - N. The same equation later was soled more recisely again using numerical methods by Howarth [], who reorted results to decimal laces. The numerical alues of f, df/dη, d 2 f/ in Table 9.1 were calculated with a ersonal comuter using fourthorder Runge-Kutta numerical integration. The elocity rofile is obtained in dimensionless form by lotting u/ ersus η, using alues from Table 9.1. The resulting rofile is sketched in Fig. 9.3b. Velocity rofiles measured exerimentally are in excellent agreement with the analytical solution. Profiles from all locations on a flat late are similar; they collase to a single rofile when lotted in nondimensional coordinates. From Table 9.1, we see that at η., u/.992. With the boundary-layer thickness,, defined as the alue of y for which u/.99, Eq. 9.7 gies ffiffiffiffiffiffiffiffiffiffiffiffi : :x ffiffiffiffiffiffiffi =νx Re x ð9:13þ The wall shear stress may be exressed as τ w μ μ ffiffiffiffiffiffiffiffiffiffiffiffi d 2 f y η

3 Table Laminar Flat-Plate Boundary Layer: Exact Solution w-21 The Function f(η) for the Laminar Boundary Layer along a Flat Plate at Zero Incidence η y νx f fu u f Then τ w :332 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρμ=x :332ρ2 ffiffiffiffiffiffiffi Re x ð9:14þ the wall shear stress coefficient, C f, is gien by C f 1 2 τ w :664 ffiffiffiffiffiffiffi ρ2 Re x ð9:1þ Each of the results for boundary-layer thickness,, wall shear stress, τ w, skin friction coefficient, C f, Eqs through 9.1, deends on the length Reynolds number, Re x, to the one-half ower. The boundary-layer thickness increases as x 1/2, the wall shear stress skin friction coefficient ary as 1/x 1/2. These results characterize the behaior of the laminar boundary layer on a flat late. Examle 9.2 LAMINAR BONDARY LAYER ON A FLAT PLATE: EXACT SOLTION se the numerical results resented in Table 9.1 to ealuate the following quantities for laminar boundary-layer flow on a flat late: (a) / (for η as η - N). (b) / at the boundary-layer edge. (c) Ratio of the sloe of a at the boundary-layer edge to the sloe of ersus x.

4 w-22 Chater 9 External Incomressible Viscous Flow Gien: Numerical solution for laminar flat-late boundary layer, Table 9.1. Find: (a) / (for η as η - N). (b) / at boundary-layer edge. (c) Ratio of the sloe of a at the boundary-layer edge to the sloe of ersus x. Solution: The dislacement thickness is defined by Eq. 9.1 as Z N 1 2 u Z 1 2 u In order to use the Blasius exact solution to ealuate this integral, ffiffiffiffiffiffiffiffiffiffiffiffi we need to conert ffiffiffiffiffiffiffiffiffiffiffiffi it from one inoling ffiffiffiffiffiffiffiffiffiffiffiffi u y to one inoling fu ( u/) η ariables. From Eq. 9.7, η y =νx; so y η νx= dη νx= Thus, Z ηmax Z νx νx ηmax ð1 2 f uþ dη ð1 2 f uþdη ð1þ Note: Corresonding to the uer limit on y in Eq. 9.1, η max N, orη max. From Eq. 9.13, ffiffiffiffiffiffiffiffiffiffiffiffi =νx so if we diide each side of Eq. 1 by each side of Eq. 9.13, we obtain (with fudf/dη) 1 Z ηmax 1 2 df dη dη Integrating gies Ealuating at η max, we obtain 1 ½η 2 f ðηþšη max 1 ð: 2 3:2833Þ :343 ß The quantity η 2 f(η) becomes constant for η. 7. Ealuating at η max 8 gies ðη Þ 1 ð8: 2 6:2792Þ :344 ß Thus, η-n is.24 ercent larger than η-. From Eq. 9.1, 1 ffiffi ν 2 x f ; so 1 ffi ν 2 x f ðη-nþ 1 2 ffiffiffiffiffiffiffi Re x f Ealuating at the boundary-layer edge (η ), we obtain 1 2 ffiffiffiffiffiffiffi Re ½ð:991Þ 2 3:2833Š :837 ffiffiffiffiffiffiffi :84 ffiffiffiffiffiffiffi x Re x Re x ß ðη Þ Thus is only.84 ercent of at Re x 1 4, only about.12 ercent of at Re x 3 1.

5 9.3 Laminar Flat-Plate Boundary Layer: Exact Solution w-23 The sloe of a at the boundary-layer edge is u :84 ffiffiffiffiffiffiffi Re x The sloe of the boundary-layer edge may be obtained from Eq. 9.13, νx ffiffiffiffiffiffiffiffiffiffiffiffi =νx so Thus, :84 d 2: rffiffiffiffi ffi d ν 1 ν 2 x21=2 2: 2: ffiffiffiffiffiffiffi x Re x :336 d ß This result indicates that the sloe of the s is about 1 3 of the sloe of the boundary layer edge the s enetrate the boundary layer, as sketched below: u This roblem illustrates use of numerical data from the Blasius solution to obtain other information on a flat late laminar boundary layer, including the result that the edge of the boundary layer is not a.

FLUID MECHANICS. Lecture 7 Exact solutions

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