Boundary Layer Similarity

One of the fundamental notions in fluid mechanics is to analyze experimental observables using dimensional analysis with the intent of finding scaling parameters that render the scaled observable from different stations along the flow to appear to be similar. These similar flow features indicate that the flow governing equations can be simplified from a set of partial differential equations to a set of ordinary differential equations thereby greatly simplifying the solution. Similarity of the velocity profile formed by fluid flow along a wall is one of those fundamental notions. For 2-D wall-bounded flows, velocity profile similarity is defined as the case where two velocity profiles taken at different stations along the flow differ only by simple scaling parameters in y and u(x,y), where y is the normal direction to the wall, x is the flow direction, and u(x,y) is the velocity parallel to the wall in the flow direction. The common approach to similarity solutions is to look for possible mathematical solutions. In the following, we take on the challenge of actually identifying the similarity scaling parameters. A proof developed by Weyburne [30] is outlined that indicates that for similarity to be present in any 2-D fluid boundary layer, the length scaling parameter must be the displacement thickness δ₁(x) and the velocity must be the velocity at the boundary layer edge u_e(x). New results presented below indicate that if similarity is present in a set of velocity profiles, then the momentum thickness δ₂(x) must also be a similarity scaling parameter. In the first two sections, we review the bounded and unbounded velocity profile cases.

The same approach to velocity profile similarity can be applied to the temperature profile case. Following the velocity profile cases, the temperature profile similarity case is briefly reviewed.

6.1 Similarity of the Boundary Layers Velocity Profile

Traditionally, the theoretical study of similarity of boundary layer flows involved the mathematical aspects of solutions to the Navier-Stokes equation, e.g. the Blasius [9] and Falkner-Skan [26] approaches. Recently, Weyburne [30] described a way to determine the identity of the similarity parameters for boundary layer flows. The approach is based on a simple concept; the area under a set of scaled velocity profile curves that show similarity behavior must be equal. This led to a new integral-based derivation (Weyburne [30]) that proved that if similarity is present in a set of bounded boundary layer velocity profiles, then the similarity velocity scaling parameter must be the velocity at the boundary layer edge u_e(x) and the similarity length scaling parameter must be the traditional displacement thickness, δ₁(x).

The bounded boundary layer proof begins by assuming a steady 2-D boundary layer flow develops along an interior channel wall (wind tunnel). The velocity along the channel wall is u(x,y), where y is the normal direction to the wall, x is the flow and wall direction. The velocity profile is defined as the velocity u(x,y) above the wall for all y at a fixed x position. The velocity profile is assumed to directly asymptote to the boundary layer edge velocity u_e(x). {Strictly speaking, the velocity will not asymptote to a constant velocity value but will peak instead. However, as we discussed in Chapter 1, wind tunnel data is limited in the y-extent so appears to plateau. Therefore, without recourse, we adopt this assumption.}

Assume a set of scaled profiles is found that show similarity when scaled with the as yet unidentified velocity scaling parameter u_s(x) and the unidentified thickness scaling parameter δ_s(x). According to Schlichting [2], boundary layer similarity is then defined as the case where the scaled velocity profile at a station x₁ is similar to the scaled profile at x₂ if

The velocity u(x,y) is written in this way to specify that the scaled velocity profile thickness comparisons are made at the equivalent y/δ_s(x) values and not at the equivalent y-values. Mathematically, a more explicit way to define the case for velocity profile similarity is to require that

Rather than starting with velocity profile similarity, we first consider the derivative of the velocity profile with respect to the scaled y-value since this result will be needed for the next step. Similarity necessarily requires that if the scaled velocity profiles are similar, then the scaled derivatives of the velocity profiles must also be similar. This, in turn, means that the scaled derivative profiles plotted versus the scaled y-value must all have equal areas under the scaled derivative curves. In mathematical terms, the area under the scaled first derivative profile curve is expressed by

Applying the standard boundary conditions, we have u(x_i,0) = 0 and u(x_i,h_i) = u_e(x_i). Using a simple variable switch, then Eq. 6.3 can be shown to reduce to

Similarity requires that a(x₁) = a(x₂). Similarity also requires that h₂/δ_s(x₂) =h₁/δ_s(x₁) but this is

easily satisfied as long as h₁ and h₂ are chosen appropriately and are located deep into the free stream but less than H/2 where H is the channel spacing. Since u_s(x) is a similarity parameter, this result necessarily requires that if similarity is present in a set of velocity profiles in the near-wall region, then u_e(x) must also be a similarity velocity scaling parameter.

This same result is also obtained from the definition equation for similarity, Eq. 6.2, by taking the limit y₁ and y₂ located deep into the free stream. In addition, previous theoretical studies by Rotta [31], Towsend [32], Castillo and George [33], and Kitsois, et. al. [34] have all identified u_e(x) as a similarity scaling parameter for 2-D turbulent boundary layer flows. Yet, there is extensive literature that has dealt with the search for other velocity scaling parameters for 2-D turbulent boundary layer flows. They have gone so far as to propose other velocity scaling parameters as being somehow superior to u_e(x). However, none of those proposing alternatives ever explain why all of these different theoretical approaches that have identified u_e(x) as a similarity scaling parameter are somehow wrong or faulty. This flawed alternative approach is exposed in more detail in Chapter 7. In any case, unless one changes the definition of similarity, or the velocity boundary conditions, the velocity at the boundary layer edge u_e(x) must be the velocity similarity scaling parameter.

With the velocity scaling parameter in hand, we next consider the area under the velocity profile curves, or in this case, the defect profiles. If similarity is present in a set of velocity profiles, then the equality condition in Eq. 6.2 is still satisfied if the constant u_e(x₁)/u_s(x₁) is subtracted from both sides. Using the similarity result from Eq. 6.4 then this means similarity must also be present in the scaled defect profiles, {u_e(x_i)-u(x_i,y)}/u_s(x_i), i=1,2. It is self-evident that if the defect profiles are similar, then the area under these scaled defect profiles plotted versus the scaled y-coordinate must be equal. The area under the scaled defect profiles, in mathematical terms, is given by

Similarity requires that b(x₁) = b(x₂). By employing a simple variable switch, Eq. 6.5 can be shown to reduce to

by recognizing that the displacement thickness, δ₁(x), is defined as

Since similarity requires that b(x₁) = b(x₂), then, combining with the Eq. 6.4 result, Eq. 6.6 necessarily means that if similarity is present in a set of velocity profiles, then the displacement thickness δ₁(x) must be a similarity length scaling parameter. This result applies to all bounded 2-D fluid flows along a flat plate.

The net result is that if similarity is present in a set of 2-D bounded velocity profiles, then the length scale must be the displacement thickness δ₁(x) and the velocity scale must be u_e(x). The results are mathematically rigorous for boundary layer flows that asymptote to u_e(x); they are only dependent on the definition of similarity, the definition of the displacement thickness, and the boundary conditions.

6.1.1 A Sufficient Condition for Similarity

Having equal areas for the profiles scaled with δ₁(x) and u_e(x) is a necessary but not a sufficient condition for the profiles to be similar. With a little more work, it is possible to develop a sufficient condition for similarity. From above, the scaled defect profiles must be similar. We also know that y_i/δ_s(x_i), i=1,2, must be equivalent (see Eq. 6.2), so it is self-evident that the scaled defect profiles multiplied by the scaled y-coordinate raised to the nth power must also be similar. In mathematical terms, the area under the scaled defect profiles multiplied by the scaled y‑coordinate raised to the nth power is equivalent to

Mathematically, it is self-evident that a sufficient condition for similarity is that the profiles at x₁ and x₂ satisfy the condition

6.1.2 Determining Similarity in Experimental Data Sets

At this point, it is not clear that all of the c_n(x_i) integrals are finite for boundary layer flows, in general. Nevertheless, this set of equations offers a new way to determine if a set of profiles displays similarity. In the literature, experimental velocity profile similarity is determined by plotting all of the profiles on a single graph and then subjectively judging whether similarity is present based on a “chi-by-eye” examination of the graphed data sets. The c_n(x_i) similarity requirement offers a statistical method for determining whether similarity is likely present in a set of velocity profiles. It is not feasible to calculate all of the c_n(x_i) values for a set of velocity profiles. However, to merely find likely datasets displaying similarity, it is only necessary to calculate and statistically compare a few n-values. Weyburne [35] used this approach to search for turbulent boundary layer datasets displaying outer-region similar-like behavior when scaled with different scaling parameter sets.

6.1.2 Momentum Thickness Similarity

The displacement thickness, δ₁(x), is not the only integral moment parameter that works as a thickness similarity parameter. Using the same type of analysis used above, Weyburne [14] pointed out that that other moment method thickness parameters, such as the second derivative mean location, μ₁(x), are also similarity scaling parameters. As we show below, it is also possible to prove that the momentum thickness, δ₂(x) must be a similarity length scale. To start, consider that in the last section we used simple logic to show that if similarity is present in the scaled velocity profiles then similarity must also be present in the scaled defect profile defined as {u_e(x)‑u(x,y)}/u_s(x). The equality condition for the scaled defect profile will still be satisfied if u(x₁,y₁)/u_s(x₁) is multiplied by both sides. Since similarity requires u(x₁,y₁)/u_s(x₁) = u(x₂,y₂)/u_s(x₂), as long as y₂ = y₁δ_s(x₂)/ δ_s(x₁), then it is also self-evident the defect profiles times u(x_i,y_i)/u_s(x_i) must be similar and that the area under these new profiles plotted versus the scaled y-coordinate must be equal. The area under these new scaled profiles, in mathematical terms, is given by

Using a simple variable switch and simple algebra, Eq. 6.10 reduces to

where the δ₂(x) is the momentum thickness is defined as

Similarity requires that e(x₁) = e(x₂). Using the result given by Eq. 6.11 combined with the result given by Eq. 6.4, then it is evident that if similarity is present in a set of scaled velocity profiles, then the momentum thickness δ₂(x) must be a length scale that results in similarity.

Since the early days of boundary layer research, the flow community needed a way to compare experimental results from different sources. It became common to categorize scaled velocity profile plots according to their Reynolds number based on the momentum thickness as the length scale, Re_θ. This approach is still the defacto standard today, especially for turbulent boundary layers where laminar to turbulent transition locations can vary greatly. The similarity result above gives a theoretical foundation for comparing velocity profiles at the same relative velocities.

6.2 Similarity of the Unbounded Boundary Layer

Weyburne [5] extended the same type of argument used above to the 2-D unbounded boundary layer case. For the unbounded boundary layer case, the boundary layer can be conveniently divided into two regions using δ_max, the location of the maximum velocity u_max, as the dividing location. The question then is the maximum peak velocity a similarity scaling parameter? For the unbounded boundary layer, the equivalent similarity length scaling parameters to go with u_max(x) as the velocity scale is the maximum displacement thickness, δ₁^m , given by

Figure 6.1 presents some experimental results for 200-meter laminar channel flow simulation case discussed in Chapter 1 using δ₁^m , δ₂^m , and u_max(x) as the scaling parameters. The δ₂^m parameter is the momentum thickness equivalent expression of Eq. 6.13. The results indicate that δ₁^m , δ₂^m , and u_max(x) appear to show good collapse of all seven profiles indicating similar-like behavior for both thickness parameters.

Although this figure shows similarity-like behavior, one must be careful. Similarity requires that the area under the curves must be equal. This means that the integral limits h₁ and h₂ locations now become important. In contrast, for the bounded boundary layer case treated above, the velocity profile is assumed to plateau to u_e(x) so choosing the integral limit values is not critical. One can arbitrarily pick a h₁ location deep in the free stream (but less than the H) and then set h₂ = δ₁(x₂) h₁/δ₁(x₁). This, therefore, ensures the calculated areas under the curves will be equal if similarity is present for the bounded case. However, for the unbounded boundary layer case, this way of selecting the integral limits no longer works since, instead of plateauing, the velocity profiles are peaking. Instead of arbitrarily picking a h₁ location and calculating h₂, the integral limits are assumed fixed to h₁=δ_max(x₁) and h₂=δ_max(x₂). For the curves in Figure 6.1, using δ_max(x) as the integral limit resulted in the calculated area under curves having a 8% standard deviation. Hence, the equal-area condition is only approximately true. It is not clear whether this difference is dependent on the simulation size, extents, etc. so it is not possible to make any definitive statements on whether unbounded laminar boundary layer flows are similar or not. More work needs to be done in this area.

It should also be mentioned that Weyburne [3] speculated that the inertial boundary layer region for unbounded laminar flows might display similar behavior. However, it was subsequently found that this was not true for unbounded laminar flow along a flat plate (Weyburne[5]).

6.3 Similarity of the Thermal Boundary Layer Profile

The similarity of the thermal boundary layer formed by fluid flowing a long a heated or cooled wall can be handled in a similar fashion as the velocity profile similarity (see Weyburne [36]). The Navier–Stokes governing equations can be extended to include the energy balance equations. The solution to this expanded set of partial differential equations for a particular flow situation is, in general, very time-consuming even with very fast computers. Under certain flow conditions, the thermal profile and the velocity profile downstream can look geometrically similar to the upstream flow, differing only by simple scaling parameters. When this happens, the set of partial differential flow governing equations can be simplified to an easily solved set of ordinary differential equations. The similarity solution for heated/cooled boundary layer flows represents one of these known solution sets.

To construct the thermal profile similarity proof, we consider a boundary layer flow along a heated/cooled inside wall of a 2-D channel with gap H. The channel half-width, H/2, is assumed to be much thicker than the maximum velocity and thermal boundary layer thickness found in the channel. Let the velocity be u(x,y) and temperature T(x,y), where y is the normal direction to the wall and x is the flow direction. Assume the flow conditions discussed in Section 6.1 hold and that the channel wall is isothermal with a temperature, T_w, and the free stream is isothermal with a temperature, T₀. The temperature profile is defined as the temperature T(x,y) taken at all y values starting from the wall moving outwards at a fixed x value. Temperature profile similarity is defined as the case where two temperature profile curves from different stations along the wall in the flow direction differ only by a scaling parameter in y and a scaling parameter of the temperature profile T(x,y).

Assume a set of temperature profiles are found that display temperature profile similarity when scaled with the height scaling parameter δ_q(x) and the temperature scaling parameter T_s. The scaled temperature profile at a station x₁ is said to be similar to the scaled profile at x₂ if

An important observation about the isothermal wall is that in the limit of y → 0 and y → H/2, the temperature ratios in Eq. 6.15 reduce to T_w /T_s(x) and T₀ /T_s(x), respectively. This means that T_s(x) must be a constant. Hence, for similarity, only the length scaling parameter δ_q(x) may vary with the flow direction.

To discover the identity of T_s and δ_q(x), the same technique that was used for the velocity profile similarity method discussed above is applied. As with the velocity case, we start by considering the first derivative of the temperature profile since this will be needed in the next step. If similarity is present in a set of temperature profiles, then it is self-evident that the properly scaled first derivative profile curves (derivative with respect to the scaled y‑coordinate) must also be similar. It is also self-evident that the area under the scaled first derivative profiles plotted against the scaled y‑coordinate must be equal. In mathematical terms, the area under the scaled first derivative profile curve is given by

where h_i is deep into the free stream above the wall but less than H/2. Using the boundary conditions and a simple variable switch, Eq. 6.16 can be shown to reduce to

Similarity requires that p(x₁) = p(x₂). Hence, the similarity scaling parameter T_s must be proportional to the temperature difference T₀ -T_w .

Now consider the identity of δ_q(x). Starting with the formal definition of similarity given by Eq. 6.14 then it is self-evident that for the profiles to be similar, the area under these scaled temperature profiles plotted versus the scaled y-coordinate must be equal. The area under the scaled temperature profile is not easily manipulated. However, Eq. 6.17 can be used to advantage. If a constant is added or subtracted from both sides of Eq. 6.14 and then integrated, the equivalence condition still holds. Subtracting T₀/(T₀ -T_w) value from both sides of Eq. 6.14 and integrating, the area, in mathematical terms, is given by

Using a simple variable switch and simple algebra, Eq. 6.18 can be shown to reduce to

where the thermal displacement thickness, δ₁^T(x) , is defined as

Similarity requires that r(x₁) = r(x₂). The importance of Eq. 6.20 in regards to similar profiles is that it means that if similarity is present in a set of thermal profiles for any 2-D boundary layer along a wall, then the thickness scaling parameter δ_q(x) must be proportional to the thermal displacement thickness δ₁^T(x). From Eq. 6.17, we showed that the similarity temperature scaling parameter T_s must be proportional to the temperature difference T₀ -T_w. The results are mathematically rigorous; they are only dependent on the definition of similarity, the definition of the thermal displacement thickness, and the boundary conditions.

To demonstrate this new result, we apply the new similarity scaling parameters to a data set from the literature. In Fig. 6.1a the results for the Pohlhausen [20] based approach to calculating the thermal profiles for laminar flow over a heated plate for a range of Prandtl numbers is presented. This figure is a re-creation of Fig. 12.9 from Schlichting [2] using a simple FORTRAN program to generate the solutions. In Fig. 6.1b, the same data is plotted using the new thermal displacement thickness scaling parameter. In this case, all nine curves are collapsed onto one another indicating thermal profile similarity is present. This is a remarkable result that provides solid support for the integral area similarity theory.

In this figure, the reduced plotting parameters are

where ν is the kinematic viscosity.