Laminar Boundary Layer Flow Dr. Om Prakash Singh Asst. Prof., IIT Mandi www.omprakashsingh.com Laminar BL BL separation (laminar or turbulent) Flow over a cylinder Transition Reference book: Convection heat transfer by Adrian Bejan Please buy this book if you use this ppt

Question theories, nothing is perfect As students and researchers, we can learn important lessons from the history of boundary layer theory. For example: No theory is perfect and forever, not even boundary layer theory. It is legal and, indeed, desirable to question any accepted theory. Any theory is better than no theory at all. It is legal to propose a new theory or a new idea in place of any accepted theory.

Fundamental problem in convective heat transfer Fluid (U, T) Plate (T0) A flat plate of temperature T0 suspended in a uniform stream of velocity U∞ and temperature T∞, as shown in fig. If this flat plate is the plate fin protruding from a heat exchanger surface into the stream that bathes it, we want to know: The net force exerted by the stream on the plate  DRAG FORCE The resistance to the transfer of heat from the plate to the stream  HEAT TRANSFER RATE

Fundamental problem in convective heat transfer The total drag force (stress x area) by the plate is The questions of friction and thermal resistance, boil down to carrying out the calculations dictated by these two equations and the total heat transfer rate (heat flux x area) Symbols τ, q, and W stand for skin friction (shear stress experienced by the wall), wall heat flux and width (W) of the wall ,  and h are the viscosity and heat transfer coefficient

Fundamental problem in convective heat transfer The fluid layer situated at y = 0+ is, in fact, stuck to the solid wall. This is the no-slip hypothesis, on which the bulk of convective heat transfer research is based; The transfer of heat from the wall to the fluid is first by pure conduction. Therefore, , we can write the statement for pure conduction through the fluid layer immediately adjacent to the wall, Note the sign convention. The heat flux q is positive when the wall releases energy into the stream. We calculate the heat transfer coefficient when the temperature distribution in the fluid near the wall is known:

Fundamental problem in convective heat transfer To summarize, the two key questions in the field of convective heat transfer, the questions of friction and thermal resistance, boil down to carrying out the calculations dictated by eqs. (2.1) and (2.2). However, eqs. (2.3) and (2.5) demonstrate that to be able to calculate F and q, we must first determine the flow and temperature fields in the vicinity of the solid wall. Thus, it is the demand for F and q that leads to the mathematical problem of solving for the flow (u,v) and temperature (T) in the fluid space outlined in Fig. 2.1.

Flow equations in convective heat transfer Modeling the flow as incompressible and of constant property, the complete mathematical statement of this problem consists of the following. Solve four 2D equations: for four unknowns ( u, v, P, T), subject to the following boundary conditions:

Boundary conditions

Concept of Boundary Layer Boundary layer scaling analysis domain:   L BL concept valid only if   L Hence, BL analysis not valid near plate front tip where   L uU , x  L, y , v from continuity equation  U (x) x L

Concept of Boundary Layer BL movie Boundary layer is a region near the wall Away from the boundary layer (how far? ) is free stream region Free stream is a flow region not affected by the obstruction and heating effect introduced by the solid object. The free stream is characterized by Let δ be the order of magnitude of the distance in which u changes from 0 at the wall to roughly U in the free stream. Thus, in the space of height δ and length L in Fig., we identify the following scales for changes in x, y, and u:

Concept of Boundary Layer Scales for change are: In the δ × L region, then, the longitudinal momentum equation (2.8) accounts for the competition between three types of forces:

Concept of Boundary Layer In (2.14) each term represents the scale of each of the five terms appearing in eq. (2.8). Since the mass continuity equation (2.7) requires that Note that the inertia terms in eq. (2.14) are both of order U2 /L (using 2.15); hence, neither can be neglected at the expense of the other. Next, if the boundary layer region δ × L is slender, such that then the last scale in eq. (2.14) is the scale most representative of the friction force in that region. Thus, neglecting the ∂2u/∂x2 term at the expense of the ∂2u/∂y2 term in the x momentum equation (2.8) yields

Concept of Boundary Layer Invoking the same scaling argument—the slenderness of the boundary layer region—the y momentum equation reduces to Equation (2.18) is not usually discussed in connection with the boundary layer analysis of specific laminar flow problems. However, it is the basis for another important result: the replacement of ∂P/∂x by a known quantity (dP/dx) in eq. (2.17).

Concept of Boundary Layer To show that To show how this is done, consider answering the following question: In a slender region δ × L, is the pressure variation in the y direction negligible compared with the pressure variation in the x direction? Intuitively, we suspect that the answer must be ‘‘yes’’ because the region of interest (δ × L) is by definition slender. In boundary layer, pressure is function of x and y; hence, total derivative Or,

Concept of Boundary Layer To show that The orders of magnitude of the two pressure gradients can be deduced from eqs. (2.17) and (2.18) by recognizing a balance between pressure forces and either friction or inertia [eq. (2.14)]. For the present argument, it is not crucial which balance we invoke as long as the same balance is invoked in both eqs. (2.17) and (2.18). For instance, the pressure ∼ friction balance in eq. (2.17) suggests that whereas the same balance in eq. (2.18) yields

Concept of Boundary Layer To show that Now, turning our attention to the right-hand side of eq. (2.20), the ratio of the second term divided by the first term is of order Note that to complete this last statement, we had to use the mass continuity scaling [eq. (2.15)] and the slenderness postulate [eq. (2.16)]. In conclusion, the last term in eq. (2.20) is less significant as the δ × L region becomes more slender,

Concept of Boundary Layer This means that inside the boundary layer, the pressure varies chiefly in the longitudinal direction; in other words, at any x, the pressure inside the boundary layer region is practically the same as the pressure immediately outside it, Making this last substitution in the x momentum equation (2.17), we finally obtain This is the boundary layer equation for momentum, and keeping in mind how it was derived, it is a statement of momentum conservation in both the x and y directions.

Concept of Boundary Layer The boundary layer equation for energy follows from eq. (2.10), where we neglect the term accounting for thermal diffusion in the x direction, We now have only three equations to solve [eqs. (2.7), (2.26), and (2.27)] for three unknowns (u,v,T). Compare this with the ‘‘four equations and four unknowns’’ problem contemplated originally. In addition, the disappearance of the ∂2/∂x2 diffusion terms from the momentum and energy equations makes this new problem solvable in a variety of ways.

Boundary layer equations Final three boundary layer equations are: Next, we begin with the most cost-effective method of solution: scale analysis on the above three equations.

Scale analysis of Boundary layer equations  T The boundary layer equations (2.26) and (2.27) are based on the significant variations in velocity and temperature in a slender region near the solid wall. This does not mean that u and T reach their free-stream values within the same distance δ. Indeed, we have the freedom to think not of one but of an infinity of slender flow regions adjacent to the wall. Let δ (velocity boundary layer) be the thickness of the region in which u varies from 0 at the wall to U in the free stream. Let δT (thermal BL) be the thickness of another slender region super imposed on the first in which T varies from T0 at the wall to T in the free stream.

Scale analysis of Boundary layer equations In general, δ = δT (for Pr = 1) The friction equation Becomes, Thus, to estimate the wall frictional shear stress, we must evaluate the extent δ of this imaginary slender wall region.

Scale analysis of Boundary layer equations Pressure drop in the direction of flow is not significant over the longitudinal length L dictated by the plate fin. With dP/dx = 0 in eq. (2.26), the boundary layer momentum equation implies that From continuity scaling, We conclude that the two inertia terms are of the same order of magnitude.

Scale analysis of Boundary layer equations Therefore, eq. (2.29) requires that BL theory fails near tip In other words, where ReL is the Reynolds number based on the longitudinal dimension of the boundary layer region, ReL = UL/ν Equation (2.31) is an important result: It states that the slenderness postulate on which the boundary layer theory is based (δ  L) is valid provided that ReL1/2  1. Thus, eq. (2.31) is a test of whether a given external flow situation lends itself to boundary layer analysis, as ReL can easily be calculated beforehand. Furthermore, even when ReL1/2  1, eq. (2.31) can be used to assess the limitations of the boundary layer analysis: For example, the boundary layer solution will fail in the tip region of length l, short enough so that ReL1/2 is not considerably greater than unity.

Scale analysis of Boundary layer equations Hence, the shear stress scaling Becomes, Therefore, the dimensionless skin friction coefficient Cf = τ/ (1/2ρU2) depends on the Reynolds number The scaling analysis on which eq. (2.32) is based assures us that the real measured or calculated) value of τ will differ from ρ U2 ReL 1/2 by only a factor of order unity.

Boundary layer equations: Exact vs. Scaling method Skin friction coefficient derived from scaling analysis Local skin friction coefficient derived from similarity method (exact) Note: the value averaged from x = 0 to any x is twice as large as the local value calculated at x. Eq. 2.33 predicts average skin friction coefficient at length L. Note that the constant term i.e. 1.328 is of the order of magnitude of unity Eq. 2.92’ derived from more laborious method compared simple scale analysis from which eq. 2.33 is derived.

Thermal boundary layer We calculate the heat transfer coefficient when the temperature distribution in the fluid near the wall is known: Hence, heat transfer can be obtained by knowing thermal boundary layer T as where T = T0 − T is the temperature variation in the region δT × L. The boundary layer energy equation (2.27) states that there is always a balance between conduction from the wall into the stream and convection (enthalpy flow) parallel to the wall:

Thermal boundary layer (thick and thin) The δT scale needed for estimating h ∼ k/δT can be determined analytically in the following two limits: Thick thermal boundary layer Thin thermal boundary layer Plot showing the relative thickness in the Thermal boundary layer versus the Velocity boundary layer (in red) for various Prandtl Numbers. For Pr =1, the two are equal.

v from continuity equation Thermal boundary layer (thick) uU , x  L, v from continuity equation Thick boundary layer, T    / T  1 U Thermal BL T  Hydrodynamic BL u,v = 0 In this limit, the δT layer is thick relative to the velocity boundary layer thickness measured at the same L. The u scale outside the velocity boundary layer (and inside the δT layer) is U. According to eq. (2.15), the v scale in the same region is v ∼ U δ/L. This means that the second term on the left side of eq. (2.35) is of order of

Thermal boundary layer (thick) in which δ/δT << 1. The second term, (vT)/δT, is therefore δ/δT times smaller than the first, (uT)/L, and the entire left side of eq. (2.35) is dominated by the scale U T/L. Hence, the convection ∼ conduction balance expressed by the energy equation (2.35) is simply U T/L ∼ (α T)/δT2, which yields (Thermal BL) (Hydrodynamic BL) where PeL = UL/α is the Peclet number. Comparing eq. (2.37) with eq. (2.31), we find the interesting result that the relative size of δT and δ depends on the Prandtl number Pr = ν/α,

Thermal boundary layer (thick) Divide, The first assumption, δT  δ , is therefore valid in the limit Pr  1, which represents the range occupied by liquid metals. The heat transfer coefficient corresponding to the low-Prandtl number limit is or, expressed as a Nusselt number Nu = hL/k

Range of Prandtl Number (Pr) for different fluids 0.001-0.03 ~ 1.0 ~ 7 5-50 50-2000 2000-105 Light organic fluids Liquid metals Gases Water Oils Glycerin The Prandtl numbers of gases are about 1, which indicates that both momentum and heat dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals (Pr  1) and very slowly in oils (Pr  1) relative to momentum. Consequently the thermal boundary layer is much thicker for liquid metals and much thinner for oils relative to the velocity boundary layer.

Thermal boundary layer Comparison: exact vs. scaling laws Exact solution from integral solution Approximate solution from scaling laws The coefficient in exact solution is order of unity. Clearly scaling analysis gives nearly correct result compared to more complicated and expensive analysis

Thermal boundary layer (thin) uU , u from continuity equation uU Thin boundary layer, T    / T  1  U Hydrodynamic BL  T Thermal BL u,v = 0 Of considerably greater interest is the class of fluids with Prandtl numbers of order 1 (e.g., air) or greater than 1 (e.g., water or oils). As shown in Fig., the thermal thickness is assumed smaller than the velocity thickness. Note that when T  , uU (geometrically similar) Hence, geometrically, it is clear that the scale of u in the δT layer is not U but

Thermal boundary layer (thin) In eq. (2.35) we note that u/L ∼ v/δT because of mass conservation, and therefore u/L ∼ α/δT2. Combining this scale relation with eq. (2.41), after using eq. (2.31) we obtain which means that

Thermal boundary layer (thin) Thus, the assumption T   is valid in the case of Pr1/3  1 fluids. The heat transfer coefficient and Nusselt number vary as where Nu = hL/k. These scaling results agree within a factor of order unity with the classical analytical results (similarity solution or integral solution) Assignment Similarity solution Integral solution

Problem 1 Consider the laminar boundary layer frictional heating of an adiabatic wall parallel to a free stream (U,T ,see fig). Modeling the flow as one with temperature-independent properties and assuming that the Blasius velocity solution holds i.e thin boundary layer assumption holds, use scaling arguments To show that the relevant boundary layer energy equation for this problem is 2. Using scale analysis, determine the wall temperature rise T as a function of U and fluid properties for two cases convection ~ heat generation balance, for what values of Pr these scaling is valid in (a) ? and conduction ~ heat generation balance and for what values Pr the scaling in (c) valid? in which case: (a) or (c) thermal boundary layer would be thick? Draw thermal and hydrodynamic boundary layers for both the cases.

Solution

Problem 2 Air g*sin y Liquid g x Consider the laminar flow of a two-dimensional liquid film on a flat wall inclined at an angle α relative to the horizontal direction (see figure). The film flow is driven by the gravitational acceleration component (g sinα) acting parallel to the wall. Attach the Cartesian system of coordinates ( x,y) and ( u,v) to the wall, such that x and u point in the flow direction. v u   In this notation, derive the terminal velocity distribution in the liquid film u(y); in other words, determine the flow in the limit where the film inertia is negligible and the x momentum equation expresses a balance between film weight and wall friction. Determine the velocity U, the undetermined free surface velocity at y = δ, where δ is the film thickness. Note that U is undetermined because the film flow rate can be varied at will by the person who pours liquid on the incline. Consider next the heat transfer from the wall to the liquid film in the case where the film and wall temperature is T0 everywhere upstream of x = 0 and where the wall temperature alone is raised to (T0 + T) downstream of x = 0. Let δT be the thermal boundary layer thickness of the thin liquid region in which the wall heating effect is felt. Using scale analysis, demonstrate that immediately downstream from x = 0 (where δT is much smaller than δ), the thermal boundary layer thickness δT scales as [(α δ x)/U]1/3 Calculate the scale of wall heat flux.

Solution

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