The New Blasius Model

The theoretical model for boundary layer fluid flow over a flat plate was developed by Blasius [9] in the early 1900s. The simplicity of the description has led to its widespread adoption in the fluid community. It has found application in wind tunnel operations as a means of operational verification. Furthermore, certain stability and transition experiments purposely use the Blasius flow as a "known" flow condition. However, this assumption of a known flow condition is flawed; the normal velocity and the normal pressure gradient that is generated in a wind tunnel do not correspond to the Blasius flow condition. In contrast, exterior laminar flow along a flat plate is shown to be naturally described as Blasius flow.

4.1 The Blasius Model Failure

Ludwig Prandtl’s [1] boundary layer concept for steady 2-D laminar flow along a wall is often depicted as starting with a zero velocity at the wall (no-slip boundary condition) which then monotonically increases above the plate until it asymptotes to the velocity at the boundary layer edge (Fig. 1.5). Paul Richard Heinrich Blasius [9], one of Prandtl’s students, developed the theoretical model corresponding to the flow for the case where the boundary layer edge velocity is just the free stream velocity u₀. This close association with the traditional conceptual model for boundary layer flow has ensured a prominent place for the Blasius theoretical model. The Blasius model appears in almost every fluid flow textbook as an introduction to boundary layer flow as well as an introduction to similarity theoretical solutions to boundary layer flows. The universal acceptance of this model appears to be partly driven by the association between the conceptual depiction and the theoretical model, partly by the simplicity of the model, and partly by the fact that wind tunnel measurements seem to confirm its existence. The universal acceptance of the model, fueled by the wind tunnel experiments, has resulted in the Blasius theoretical model being routinely used as a way to verify proper wind tunnel configuration/operation (e.g. Jovanović, et. al. [21]) as well as a way to verify computer flow simulation computational engines (e.g. Ghia, et. al. [22]).

The near-universal acceptance of the theory is based, in part, on wind tunnel experiments beginning with Nikuradse [23] experimental wind tunnel results featured in Schlichting [2] seminal book on boundary layer theory (see Schlichting’s Fig. 7.9). This result would appear to be a powerful confirmation of the Blasius theory, in particular, and the boundary layer concept in general. However, this apparent experimental confirmation needs to be re-evaluated. Recently Weyburne [3-5] set up a series of computer simulation experiments that, in part, tried to verify the existence of Blasius type flow in a 2-D channel with typical wind tunnel dimensions (1-meter high by 8-meter long). It was found that parallel channel wall-flow did not produce Blasius-type flow for 2-D channel airflow. This result appears to correspond to actual wind tunnel experiments and is not unexpected since the pressure gradient in the flow direction will not be zero under these conditions. Nikuradse [23] results, according to Schlichting [2], employed a pressure gradient cancellation technique. One standard wind tunnel technique to induce a zero-pressure gradient along the tunnel plate and thereby generate Blasius profiles is to slightly tilt or modify the top tunnel plate to counter the built-in pressure gradient formed by the wind tunnel walls (see, for example, Jovanović, et. al. [21]). And computer simulations of a wedge-shaped channel do show that it is possible to produce u(x,y) velocity profiles that behaved similarly when scaled with the Blasius scaling parameters (Weyburne [4,5]). However, the same simulations indicated that the normal velocity profiles and the normal pressure gradient profiles do not show similarity using the Blasius scaling parameters. Overall, it was concluded that the small angle-wedge flow used to induce a zero-pressure gradient (ZPG) value at the boundary layer edge is not well represented by the Blasius theoretical model. Note that in typical wind tunnel experiments, neither the normal velocity profiles nor the normal pressure gradient profiles are usually measured. Hence, this experimental discrepancy has never been noticed. In any case, the simulation results indicate that the assumption that the Blasius theoretical model describes steady 2-D laminar flow along a tilted flat plate channel in a wind tunnel is not correct.

It is not only experimental verification problems that cast doubt on the Blasius theoretical model. A closer look at the Blasius model reveals a disturbing theoretical discrepancy that one would not expect for fluid flow along a flat plate. The problem is that the Blasius model does not permit momentum conservation since the velocity asymptotes directly to the free stream velocity. The displaced momentum caused by the presence of the boundary layer just disappears in this model. As we showed in Chapter 1, the consequence of this displaced momentum in the boundary layer region shows up as a velocity overshoot, a peak, near the boundary layer edge. This behavior is not observed for the Blasius model as it is normally applied to boundary layer flows.

Another implied problem with the Blasius solution is that the calculated normal to the wall velocity and the normal to the wall pressure gradient are both finite at an infinite distance from the wall. This has resulted in several incorrect attempts to explain this infinite issue (see, for example,Lewins [24] and Panton [28], p. 556 ). We say implied problem since the literature has failed to emphasize that the Blasius equations are only supposed to apply to the boundary layer region according to Prandtl’s [1] original boundary layer hypothesis.

4.2 The New Blasius Theoretical Model

The theoretical problems just mentioned, coupled with the wedge channel computer simulation experiments indicate that the Blasius [9] theoretical model needs to be re-evaluated. Recently, Weyburne [5] did just that by reapplying the Blasius theoretical model in a context that removes all of the just-mentioned problems. The key is not the theoretical model but the conceptual model for the boundary layer. Applying the new unbounded boundary layer conceptual model, we assume that the “unbounded” boundary layer is divided into a viscous near-wall region, then a mostly inviscid peak region, and finally the inviscid outer region. The Blasius theoretical model corresponds to the near-wall viscous region. The viscous Blasius region of this new unbounded boundary layer consists of the region from the wall up to the maximum velocity location, δ_max, near the viscous boundary layer edge. The mostly inviscid region consists of the region past the maximum peak velocity location out to δ_i (it is mostly inviscid since the viscous forces are proportional to the velocity difference u_max-u₀ versus u₀ in the near wall region). Prandtl’s [1] original inviscid region now becomes the region past δ_i. The maximum velocity location is chosen as the dividing point for easy conceptualization. We could just as easily have chosen the viscous boundary layer thickness, δ_v, location instead. As we showed earlier, it appears that the δ_max and δ_v locations are nearly the same for both laminar (see Chapter 1.2.2) and turbulent (see Figure 2.6) boundary layer flows.

The association between the unbounded boundary layer near-wall region and the Blasius theoretical model has distinct theoretical advantages. The original Prandtl concept provides the physical and a logical justification for applying the Blasius model to just the viscous near-wall region. Applying the Blasius model to just the inner region eliminates the normal to the wall velocity and the normal to the wall pressure gradient infinite extent problem. The new division also solves the lack of a velocity boundary layer peak that was not incorporated in the original model. The net result is that the new Blasius theoretical model has a solid theoretical basis.

While the theoretical advantages are satisfying, it is the experimental results that clinch it for the new Blasius model. To our knowledge, no one has attempted to experimentally measure boundary layer velocity profiles and the y-pressure gradient on a thin flat plate in an exterior environment. Fortunately, computer simulations of exterior flow along a wall can provide the missing information. Weyburne [5] simulated airflow along an 8-meter-long channel with the gap initially set at 1‑meter to mimic a standard wind tunnel and then increased until the flow behaved as an exterior flow. As evidenced by the normal velocity profile behavior (see Fig. 1.7), exterior-like flow does not occur until the gap is increased to 200-meters for laminar flows with a Reynolds number Re_x=5x10⁵ (the critical laminar-turbulent transition Reynolds number)! The simulations indicate that this large gap, zero-incident angle unbounded boundary layer case is naturally in a ZPG flow condition (see Fig. 1.10b). This makes the “unbounded” boundary layer 200-meter simulation a good candidate for testing the Blasius theoretical model along a thin flat plate. The results are shown in figures Figs. 4.1-4.3. For comparison purposes, we also show the simulated wind-tunnel-like 1-meter results.

Figs. 4.1-4.3 show the scaled u(x,y) velocity profiles, the scaled v(x,y) velocity profiles, and the scaled y-pressure gradient profile for the 1‑meter gap tilted wedge channel flow and for the 200-meter gap channel flow case at seven locations along the plate. The tilt angle for the 1-meter simulation was adjusted until the scaled u(x,y) velocity profiles along the plate could be made to appear almost identical along the plate using the Blasius scaling parameters (Fig. 4.1a). Notice that the tilted 1‑meter normal velocity and normal pressure gradients displayed in Figs. 4.2a and 4.3a do not show similar-like behavior. On the other hand, the 200-meter velocities and normal pressure gradients show good collapse to the Blasius theoretical result except for the six- and seven‑meter y‑pressure gradient result. Strictly speaking, the peaking means similarity is not present but, because the velocity peaks are very small, about 0.1% of u₀, the 200-meter gap channel flow results have similar-like behavior. In any case, the similarity-like behavior in Figs. 4.1b, 4.2b, and 4.3b, combined with the dP/dx result from Fig. 1.10b, it is clear that the experimental results for the unbounded boundary layer (200‑meter) show good correspondence to the Blasius theoretical model.

Figs. 4.1-4.3 show the scaled u(x,y) velocity profiles, the scaled v(x,y) velocity profiles, and the scaled y-pressure gradient profile for the 1‑meter gap tilted wedge channel flow and for the 200-meter gap channel flow case at seven locations along the plate. The tilt angle for the 1-meter simulation was adjusted until the scaled u(x,y) velocity profiles could be made to appear almost identical along the plate using the Blasius scaling parameters (Fig. 4.1a). Notice that the tilted 1‑meter normal velocity and normal pressure gradients displayed in Figs. 4.2a and 4.3a do not show similar-like behavior. On the other hand, the 200-meter velocities and normal pressure gradients show good collapse to the Blasius theoretical result except for the six- and seven‑meter y‑pressure gradient result. Strictly speaking, the peaking means similarity is not present but, because the velocity peaks are very small, about 0.1% of u₀, the 200-meter gap channel flow results have similar-like behavior. In any case, the similarity-like behavior in Figs. 4.1b, 4.2b, and 4.3b, combined with the dP/dx result from Fig. 1.10b, it is clear that the experimental results for the unbounded boundary layer (200‑meter) show good correspondence to the Blasius theoretical model.

It should be pointed out that the 1-meter case result shown above is different from the Air Force Tech Report [5] result. Both are 1-meter cases that have the outlet set to 1.026-meter (the optimized ZPG wedge forming value) but in the figures above, the bottom plate is kept flat and the top plate was adjusted. In the Air Force Tech Report, both the top and bottom plates were adjusted equally to give a 1.026-meter output. In the Air Force Tech Report result, the normal velocities are actually negative; the normal flow is actually toward the plate. The normal pressure gradients also showed the non-similar behavior shown above. The bottom line is that wind-tunnel-like flow does not correspond to Blasius similarity flow no matter how it is setup. This behavior confirms the general observation that ZPG flows are very difficult to establish in a wind tunnel and that small differences in the ZPG conditions could have unintended effects on certain stability and transition experiments.

4.3 Experimental Verification of the Blasius Pressure Gradient Assumption

The Blasius theoretical model assumes the x-pressure gradient is zero in the x-momentum equation. In Fig. 4.4 the simulation results for the x and y pressure gradients at the mid-point of the plate for the 1-meter and 200-meter cases are displayed. It is evident that the 200-meter exterior-like case shows that the x-pressure gradient in the boundary layer region is almost zero, and smaller than the y-pressure gradient (if it was exactly zero, there would be no flow). This is not true for the 1-meter wedge flow case. This provides further experimental proof that the flow along an exterior thin flat plate is well described by the Blasius theoretical model. At the same time, it indicates that the wedge channel flow typical in a wind tunnel is not well approximated by the Blasius formulation.