Similarity solutions
From ThermalFluidsPedia
The form of the velocity, temperature, and concentration profiles for flow over a flat plate are presented in the figure to the right, and are based on qualitative measurements. These presentations indicate the possibility that these profiles are geometrically similar to each other along the flow direction for dependent variables such as velocity, temperature, and concentration, if the coordinates are stretched properly. For example, the velocity profiles are geometrically similar along the flow in the xdirection, differing only by a stretching factor (similarity factor) if the coordinates are properly sketched^{[1]}.
As noted before, not all boundary layer flow configurations have similar geometric profiles (similarity solution), but some do ^{[2]}, especially for more conventional geometries and boundary conditions. The steady, twodimensional, laminar, momentum boundary layer equation is a second order, nonlinear, partial differential equation. The twodimensional, steady, laminar flow, energy, and species boundary layer equations are linear, second order, partial differential equations for most constant properties, and/or for decoupled heat and mass transfer problems.
If a similarity solution exists for a given situation, a mathematical transformation of coordinate systems can be performed to reflect this fact. A similarity technique converts conservation partial differential equations into ordinary differential equations and therefore, makes the solution much simpler. However, the analytical solutions may still require numerical integrations.
The conservation equations are uncoupled when each equation and its boundary condition can be solved independently of one another, except for continuity and momentum, which must be solved simultaneously. Coupled transport phenomena can occur in some applications because of coupled conservation governing equations and/or boundary conditions for mass diffusion, momentum, or heat transfer.
Contents 
Uncoupled Mass, Momentum, and Heat Transfer Problems
Similarity Solution for Flow over a Flat Plate
We first apply the similarity solution to the classic problem of forced convective flow over a flat plate with constant free stream velocity, where mass, momentum, and heat transfer equations are uncoupled. Next, similarity solutions of control equations with appropriate boundary conditions of above figure are presented below. The viscous dissipation term is neglected for simplicity even though it is possible to obtain a similarity solution with viscous dissipation for this simple configuration. Assuming geometrically similar velocity profiles are possible along the flow over a flat plate using the following nondimensional variables ^{[3]}^{[4]}^{[2]}^{[5]}:
 (1) 
 (2) 
where η is a nondimensional independent variable (function of both x and y) that fulfills the similarity requirement. Mass and momentum boundary layer equations for flow over a flat plate (refer to Boundary layer approximations) can be converted to a new coordinate system η, f, and g by using the notation , and.
 (3) 
 (4) 
 (5) 
Substituting eqs. (3)(5) into mass and momentum boundary layer equations for flow over a flat plate yields:
 (6) 
 (7) 
Combining eqs. (6) and (7) to eliminate v and separating variables gives:
 (8) 
Considering that both η and x are independent variables, the left side of eq. (8) is a function of η, and the right side is a function of x, then each side must be a constant.
 (9) 
Integrating eq.(9) gives:
 (10) 
At the leading edge of the flat plate (x = 0), the boundary condition is which can be satisfied only at . In other words, η must be infinite when x = 0. It follows from eq. (2) that g(0) must be infinite and consequently, c_{2} in eq. (10) must be zero. Therefore
 (11) 
 (12) 
Since c_{1} is constant, the axial velocity is of the following form:
 (13) 
From eq. (8)
 (14) 
Integrating eq. (14) gives:
 (15) 
The following boundary conditions are used to evaluate constants c_{1} and c_{3}:
 (16) 
Using eqs. (15) and (16):
 (17) 
then, using eqs. (15) and (4.17):

therefore,
 (18) 
Let the function represent the nondimensional axial velocity, :
 (19) 
Then, using eq. (19)
 (20) 
Substituting eq.(20) in eq.(18) gives:
 (21) 
or
 (22) 
From eq.(22) we conclude that c_{1}U_{∞} is a constant, nondimensional variable. Let , eq. (22) becomes
 (23) 
where
 (24) 
Equation (23) is known as the Blasius equation and is a nonlinear, third order, ordinary differential equation, which requires three boundary conditions.
 (25) 
Equation (23) can be numerically integrated using boundary conditions given by eq. (25). Blasius clearly integrated eq. (23) by hand without using a computer ^{[6]}. The numerical results using the shooting method are presented in the figure to the right, which shows that both u and vas functions of the independent variable η. A number of conclusions can be made from the numerical results presented in the figure to the right.
1. The edge of the momentum boundary layer is approximately at η=5.
2. The vertical velocity component v is nonzero at the edge of the boundary layer at η=5, even though the free stream flow is parallel to the plate.
3. The momentum boundary layer thickness is approximately
 (26) 
where
Using the definition of shear stress at the wall, τ_{w}, and the skin friction coefficient, c_{f}, we obtain:
 (27) 
Rearranging the above equation in terms of gives:
 (28) 
 (29) 
For the purposes of design and comparison with experimental measurements, it is more practical to obtain the average value of the skin friction, c_{f}, which is usually determined by
 (30) 
where W is the width of the plate. From eq.(29),
 (31) 
It should be emphasized that the results presented above are for laminar flow parallel to a flat plate with constant free stream velocity. The approach to obtain the velocity distribution presented above can be extended to obtain the temperature distribution for laminar boundary layer flow over a flat plate with constant properties, constant free stream velocity, and constant temperature. The constant property assumption is important to decouple the momentum and energy equations in this configuration in order to obtain velocity first and temperature second. The nondimensional temperature, θ, is defined as:
 (32) 
The nondimensional form of the energy equation (neglecting viscous dissipation) and boundary conditions and using the variables η and noted in eqs. (24) and (19) are as shown below.
 (33) 
 (34) 
 (35) 
with the notation that and . Equation (33) can be directly integrated:
 (36) 
 (37) 
 (38) 
Applying boundary conditions (34) and (35) gives
 (39) 
Therefore, we obtain the following equation for the dimensionless temperature, θ:
 (40) 
The energy equation (33) is a second order, linear, ordinary differential equation. The dimensionless temperature, θ, is a function of η and Pr. Equation (40) can be integrated numerically for a given Pr. The numerical results of θ as a function of η for different Pr values are presented in the figure to the right.
The following conclusions can be made upon reviewing the results of the momentum and thermal boundary layer equations, as presented in the above two figures.
1. Temperature profile is identical to velocity for Pr = 1.
2. Temperature profile has strong dependence on the Prandtl number.
3. For Pr<1, the thermal boundary layer thickness is greater than the momentum boundary layer thickness, δ_{T} > δ.
4. For Pr>1 the thermal boundary layer thickness is less than the momentum boundary layer thickness, δ_{T} < δ.
The local heat transfer coefficient can be easily obtained from the numerical results of eq. (40) for a given Prandtl number by using Fourier’s law of heat conduction.
 (41) 
The local Nusselt number is presented in terms of dimensionless variables as follows:
 (42) 
or by using eq. (4.40):
 (43) 
The numerical results can be approximated for the range of Prandtl numbers from 0.5 to 15 by the following equation:
 (44) 
The ratio of thermal boundary layer thickness, δ_{T}, to momentum boundary layer thickness, δ, for flow over a flat plate can also be approximated over this range of Prandtl numbers according to the relationship below:
 (45) 
For both design purposes and comparison with experimental measurement it is more convenient to calculate the average or mean heat transfer coefficient, .
 (46) 
From above, the mean heat transfer coefficient and mean Nusselt number become:
 (47) 
 (48) 
This shows that the average heat transfer coefficient is twice the local heat transfer coefficient, just as the mean skin friction coefficient is twice the local skin friction coefficient for flow over a flat plate. It should be noted that these conclusions concerning the relation between the local and average transfer coefficients can not be extended to other configurations.
Similarity Solution for Flow over a Wedge
Similarity solutions also exist for some other conventional geometries with simple boundary conditions. Some of these cases are discussed below. For flow over a wedge, as shown in figure below, the free stream velocity varies according to potential (inviscid) flow theory:
 (49) 
where U is the free stream velocity at the outer wedge surface and m is related to the wedge angle β by:
 (50) 
• When β (measured in radians) is positive, the free stream velocity increases along the wedge surface.
• When β is negative, the free stream velocity decreases along the wedge surface.
• β = 0 corresponds to flow parallel to a flat plate.
• β = π corresponds to flow perpendicular to the walls.
U_{∞} is the oncoming free stream velocity, which is constant, as shown in figure to the right.
Equation (49) can be also applied near the leading edge of a blunt object. For example, the free stream velocity on the surface of a cylinder and a sphere near a stagnation point can be predicted by potential flow theory according to the equations below^{[7]}:
(cylinder)  (51) 
(sphere)  (52) 
where R is the radius of the cylinder or sphere, U_{∞} is the oncoming velocity and U is the free stream velocity just outside the boundary layer. The pressure gradient term is related to free stream velocity by potential flow theory. This can be done using the inviscid momentum equation (μ = 0)
 (53) 
The two dimensional, steady, constant property boundary layer equations for mass, momentum, and energy of flow over a wedge neglecting viscous dissipation are:
Continuity equation:
 (54) 
Momentum (xdirection):
 (55) 
Energy equation:
 (56) 
The above equations can be easily obtained by beginning with the general equations and making appropriate boundary layer approximations using scaling and order of magnitude analysis. Using eq. (53) for the pressure and comparable dimensionless similarity parameters to flow over a flat plate, we get the following ordinary differential equation for momentum and energy for constant wall temperature with no blowing or suction at the wall.
 (57) 
 (58) 
The boundary conditions for velocity and temperature are the same as in the case of flow over a flat plate. Assuming that heating starts at the leading edge, the ordinary differential equations (57) and (58) for dimensionless velocity and temperature can be solved using the given boundary condition. The friction coefficient and Nusselt number can be calculated from the numerical results by the following equations:
 (59) 
 (60) 
where the constant A depends on m and Pr. Solutions of eqs. (57) and (58) for m > 0 are unique. For m < 0 two groups of solutions exist for a limited range of m values corresponding to negative β. These two groups correspond to a decelerating mainstream to the point of incipient separation or laminar boundary layer flows after separation. Falkner and Skan ^{[8]} originally developed the similarity solution for flow over a wedge with U_{∞} = cx^{m} and an impermeable wall. The numerical results for the friction coefficient for selected values of m and β are presented in the table below. m = β = 0 corresponds to the case of flow parallel to a flat plate with constant free stream values (Blasius Solution).
Table The local friction coefficient for laminar boundary layer flow over a wedge, with U = cx^{m} and an impermeable wall
β  m  
2π π π/2 π/5 0 0.14 0.18 0.199  ∞ 1.0 0.333 0.111 0 0.0654 0.0826 0.0904  ∞ 1.233 Stagnation 0.757 0.512 0.332 Flat plate 0.164 0.082 0 Separation 
Eckert ^{[9]} was the first to obtain the numerical results for eqs. (57) and (58) for constant free stream and wall temperature as presented in the following table.
Table 2 Local values of Nu_{x}Re_{x}^{1/2} for laminar boundary layer flow over a wedge with constant wall and free stream temperature and an impermeable wall with U_{∞} = cx^{m}
A  B  Pr  
1  2  3  4  5  
0.512 0 π/5 π/2 π 8π/5  0.0753 0 0.111 0.333 1.0 4  0.242 0.292 0.331 0.384 0.496 0.813  0.253 0.307 0.348 0.403 0.523 0.858  0.272 0.332 0.378 0.440 0.570 0.938  0.45 0.585 0.669 0.792 1.043 1.736  0.570 0.730 0.851 1.013 1.344 2.236 
In similarity solutions for both flow over a flat plate and over a wedge, it is assumed that there is no blowing or suction at the wall. It can be shown that similarity solutions with blowing or suction exist only if the vertical velocity at the wall changes along the flow according to the following equation:
 (61) 
where vw is the blowing or suction velocity. Clearly this requirement restricts the use of the similarity solution to special cases that satisfy eq. (61).
Coupled Mass, Momentum, and Heat Transfer Problems
Coupled transport phenomena occur frequently in problems associated with evaporation, sublimation, absorption, or combustion. Sublimation over a flat plate can find its application in an analogy between heat and mass transfer and will be used here as an example to show the methodology used in solving these coupled problems.
When the pressure and temperature of ice are above the triple point and the ice is then heated, melting occurs. However, when the ice is exposed to moist air with a partial pressure of water below its triple point pressure, heating of the ice will result in a phase change from ice directly to vapor without going through the liquid phase. Spacecrafts and space suits can reject heat by sublimating ice into the vacuum of space. Another application for sublimation of ice is the preparation of specimens using freezedrying for a scanning electronic microscope (SEM) or a transmission electronic microscope (TEM). This type of phase change is referred to as sublimation. The opposite process is deposition, which describes the process of vapor changing directly to solid without going through condensation. The phasechange processes related to solids can be illustrated by a phase diagram in to the right. Sublimation and deposition will be the subjects of this subsection. When a subcooled solid is exposed to its superheated vapor, as shown in second figure (a), the vapor phase temperature is above the temperature of the solidvapor interface and the temperature of the solid is below the interfacial temperature. The boundary condition at the solidvapor interface is
 (62) 
where h_{δ} is the convective heat transfer coefficient at the solidvapor interface, h_{sv} is the latent heat of sublimation, and δ is the thickness of the sublimated or deposited material. The interfacial velocity dδ/dt in eq. (62) can be either positive or negative, depending on the direction of the overall heat flux at the interface. While a negative interfacial velocity signifies sublimation, a positive interfacial velocity signifies deposition. When the vapor phase is superheated, as shown in second figure(a), the solidvapor interface is usually smooth and stable.
In another possible scenario, as shown in (b) of the figure to the right, the solid temperature is above the interfacial temperature and the vapor phase is supercooled. The interfacial energy balance for this case can still be described by eq. (62). Depending on the degrees of superheating in the solid phase and supercooling in the vapor phase (the relative magnitude of the first and second terms in eq. (62) both sublimation and deposition are possible. During sublimation, a smooth and stable interface can be obtained. During deposition, on the other hand, the interface is dendritic and unstable, because supercooled vapor is not stable. The solid formed by deposition of supercooled vapor has a porous structure. During sublimation or deposition, the latent heat of sublimation can be supplied from or absorbed by either the solid phase or the vapor phase, depending on the temperature distributions in both phases. Naphthalene sublimation is also a case whereby a heat transfer coefficient can be obtained through the measurement of a mass transfer coefficient and the analogy between heat and mass transfer ^{[10]}. The significant advantages of this method include its high accuracy and the simplicity of the experimental apparatus. In addition, the local heat transfer coefficient can be obtained by measuring the local sublimed depth of the specimen.
Figure to the right shows the physical model of a sublimation problem, where a flat plate is coated with a layer of sublimable material and is subject to constant heat flux underneath ^{[11]}. A gas with the ambient temperature and mass fraction of sublimable material flows over the flat plate at a velocity of U_{∞}. The heat flux applied from the bottom of the flat plate will be divided into two parts: one part is used to supply the latent heat of sublimation, and the other is transferred to the gas through convection. The sublimated vapor is injected into the boundary layer and is removed by the gas flow.
The following assumptions are made in order to solve the problem: 1. The flat plate is very thin, so the thermal resistance of the flat plate can be neglected. 2. The gas is incompressible, with no internal heat source in the gas. 3. The sublimation problem is twodimensional steady state.
The governing equations for mass, momentum, energy, and species of the problem are
 (63) 
 (64) 
 (65) 
 (66) 
A no slip condition at the surface of the flat plat requires that u = 0 at y = 0. For a binary mixture that contains the vapor, sublimable substance, and gas, the molar flux of the sublimable substance at the surface of the flat plate is
 (67) 
Since the mass fraction of the sublimable substance in the mixture is very low, i.e., ω1, the mass flux at the wall can be simplified to
 (68) 
Sublimation at the surface causes a normal blowing velocity,. The normal velocity at the surface of the flat plate is therefore
 (69) 
The energy balance at the surface of the flat plate is
 (70) 
Another reasonable, practical, representable boundary condition at the surface of the flat plate emerges by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature. The mass fraction and the temperature at the surface of the flat plate have the following relationship ^{[12]}:
 (71) 
where a and b are constants that depend on the sublimable material and its temperature. As the boundary conditions are
, ,  (72) 
Introducing the stream function ψ,
 (73) 
the continuity equation (63) is automatically satisfied, and the momentum equation in terms of the stream function becomes
 (74) 
Similarity solutions for eq. (74) do not exist unless the injection velocity vw is proportional to x1/2, and the incoming mass fraction of the sublimable substance, ω_{∞}, is equal to the saturation mass fraction corresponding to the incoming temperature, T_{∞} ^{[12]}^{[11]}. The governing equations cannot be reduced to ordinary differential equations. The local nonsimilarity solution proposed by Zhang et al. ^{[11]} will be presented here.
Defining the following similarity variables:
 (75) 
eqs. (74) and (65) – (66) become
 (76) 
 (77) 
 (78) 
where prime ' represents partial derivative with respect to η, and all upper case variables represent partial derivatives of primary similarity variables with respect to ξ.
 (79) 
It can be seen from eqs. (76) – (78) that the similarity solution exists only if F = Θ = Φ = 0. In order to use eqs. (76) – (78) to obtain a solution for the sublimation problem, supplemental equations for F, Θ, and Φ must be obtained. Taking partial derivatives of eqs. (76) – (78) with respect to ξ and neglecting the higher order term, one obtains
 (80) 
 (81) 
 (82) 
The boundary conditions of eqs. (76) – (78) and eqs. (79) – (81) are
 (83) 
 (84) 
 (85) 
 (86) 
 (87) 
 (88) 
 (89) 
 (90) 
 (91) 
 (92) 
 (93) 
 (94) 
 (95) 
 (96) 
where
 (97) 
reflects the effect of injection velocity at the surface due to sublimation, and
 (98) 
represents the effect of the mass fraction of the sublimable substance in the incoming flow. is saturation mass fraction corresponding to the incoming temperature:
 (99) 
The set of ordinary differential equations (76) – (78) and (79) – (81) with boundary conditions specified by eqs. (83) – (96) are boundary value problems that can be solved using a shooting method ^{[11]}.
Once the converged solution is obtained, the local Nusselt number based on the total heat flux at the bottom of the flat plate is
 (100) 
and the Nusselt number based on convective heat transfer is
 (101) 
The Sherwood number is
 (102) 
References
 ↑ Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
 ↑ ^{2.0} ^{2.1} Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ.
 ↑ Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGrawHill, New York, NY
 ↑ Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, Hoboken, NJ.
 ↑ Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
 ↑ Blasius, H., 1908, Grengschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phy., Vol. 56 p.4, Also NACA TM 1256
 ↑ Schlichting, H. and Gersteu, K., 2000, Boundary layer theory, 8th enlarged and revised ed., SpringerVerlag New York
 ↑ Falkner, V.M., and Skan, S.W., 1931, Some Appropriate Solutions of the Boundary Layer Equations, Phil. May 12, pp. 865896
 ↑ Eckert, E.R.G., 1942, Die Berchnung des Wärmeüberganges in der Laminaren Grenzschicht um strömter körper, VDI – Forschungsheft, vol. 416 pp. 124
 ↑ Eckert, E.R.G., and Goldstein, R.J., 1976, Measurement in Heat Transfer, McGrawHill, New York, NY
 ↑ ^{11.0} ^{11.1} ^{11.2} ^{11.3} Zhang, Y., Chen, Z.Q., and Chen, M., 1996, “Local NonSimilarity Solutions of Coupled Heat and Mass Transfer of a Flat Plate with Uniform Heat Flux in a Laminar Parallel Flow,” Journal of Thermal Science, Vol. 5, pp. 112116
 ↑ ^{12.0} ^{12.1} Kurosaki, Y., 1974, “Coupled Heat and Mass Transfer of a Flat Plate with Uniform Heat Flux in a Laminar Parallel Flow,” Journal of the Japan Society of Mechanical Engineers, part B, Vol. 40, pp. 10661072