Entrainment and boundarylayer separation: a modeling history
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Abstract
For many decades since Prandtl’s presentation of the boundarylayer equations, all attempts to calculate separated boundary layers ended up in a breakdown of the numerical algorithm. It was not until the late 1970s that the first successful calculations were reported. The paper describes the history and philosophy of the steps that led to understanding where the breakdown originates and how to circumvent it. Especially, the role of Head’s entrainment will be highlighted.
Keywords
Boundarylayer flow Entrainment Flow separation Goldstein singularity Quasisimultaneous coupling Turbulent flow Viscous–inviscid interaction1 Prologue
2 Prandtl’s “Grenzschicht”
In his landmark presentation [4] at the 3rd International Congress of Mathematicians in Heidelberg (1904), Prandtl sketched in his first figure, reproduced in Fig. 2, how the velocity profile close to the wall shows the retarding influence of a solid wall. This event was the actual birth of boundarylayer theory, as we know it nowadays. For a modern account, the reader is referred to another ‘classic’: the monograph [6] by Herman Schlichting (one of Prandtl’s students). Overview papers of boundarylayer theory can be found at several places in the literature, e.g., in the Annual Review of Fluid Mechanics [7, 8, 9, 10].
Prandtl’s second figure, also reproduced in Fig. 2, shows a sketch of streamlines in a boundary layer that separates from a solid wall. Little did Prandtl know that it would take three quarters of a century before such a separated boundary layer could be calculated as the solution of ‘his’ boundarylayer equation (1). The current paper describes the essential steps towards this result.
2.1 The ‘invisible’ boundary layer
The red area in Fig. 3 represents the supersonic (i.e., low pressure) part of the flow, which is terminated by a shock wave. The green color, visible, e.g., near the leading and trailing edges, corresponds with low flow velocities. There is also slow flow between the supersonic region and the airfoil surface, but the boundary layer is thinner than one pixel of the original picture. Yet, this ‘invisible’ boundary layer is responsible for a large decrease of the supersonic region, creating a significant reduction of the aerodynamic lift.
In spite of being very thin, a turbulent boundary layer can display much flow details. Van Dyke’s beautiful Album of Fluid Motion [13] shows many snapshots that clearly reveal the intermittent character of a turbulent shear layer, with ‘clouds’ of turbulent flow penetrating in the laminar outer flow. From these pictures, one can infer the tremendous challenge to model this type of flow.
3 Early calculation methods

\(\displaystyle {\delta ^*\equiv \int _0^\delta \left( 1\frac{u}{U}\right) \,\mathrm{d}y}\) displacement thickness effective shape of body with equal inviscid mass transport past \(y=\delta ^*\)

\(\displaystyle {\theta \equiv \int _0^\delta \frac{u}{U}\left( 1\frac{u}{U}\right) \,\mathrm{d}y}\) momentum thickness equal to inviscid momentum transport past \(y=\theta +\delta ^*\); directly related to drag

\(\displaystyle {H \equiv \frac{\delta ^*}{\theta } \ (\ge 1)}\) shape factor turbulent separated flow corresponds roughly with \(H >\approx 3\)

\(\displaystyle {c_f \equiv \frac{2\nu }{U^2} \left. \frac{\partial u}{\partial y}\right _\mathrm{wall}}\) skin friction coefficient.
Equation (2) was solved by Ernst Pohlhausen [15] (yet another of Prandtl’s students) by assuming a polynomial velocity profile with unknown coefficients plus some additional conditions on the polynomial. From their definition, the above quantities can then be expressed in the unknown polynomial coefficients and substituted in Von Kármán’s equation. Schlichting’s monograph [6] (until the 7th edition) gives a detailed description of this approach.
It appeared impossible, however, to compute separated boundary layers in this way. All calculations ended up in diverging or singular results. In 1948, a thorough analysis of this breakdown was presented by Sydney Goldstein in another landmark paper [16]. He proposed a number of possible causes for this failure, but the computational power to deeper investigate these options was not available in those days. It would take three decades before further research into this issue was possible along computational lines, eventually revealing that Goldstein had been close. Prandtl, who passed away in 1953, never has seen the outcome as we understand it nowadays.
4 M.R. Head and his ideas on entrainment
In his report from September 1958 for the Aeronautical Research Council [19], Head presents these data in a large number of graphs featuring several variables (see Fig. 6). His hope was to find some ‘magic’ pattern between the variables which could have universal value. Apparently disappointed by what he found, and guided by his own physical intuition, he decided to come up with a new variable. He proposed that the entrainment of the boundary layer would play an important and, above all, universal role.
Head fitted the assumed functional relations to experimental data [20, 21]. Figure 8, copied from [19], shows these relations together with the used measurement data. Observe that the H–\(H_{\delta \delta ^*}\) curve in Fig. 8(right) is only drawn up to \(H=3\), where (roughly) flow separation sets in. Also, observe that the curve runs more or less horizontal there. Yet it would take another two decades before the consequences thereof were realized. Head was still not able to compute separated boundary layers!
5 A minimum enters the scene
In the second half of the 1960s, further light was shed on the numerical breakdown at separation. Catherall and Mangler [22], in 1966, were the first to continue the calculations (slightly) past the point of flow separation. However, inside the region of reversed flow, once more they ran into a breakdown of the calculations. In 1969, Stewartson and Williams [23] presented an asymptotic view of the flow field near separation, introducing a socalled tripledeck structure around the singularity. Their theory gave insight into the physical flow of information near a separation point; see also [1, 9, 24]. One decade later, the last pieces of the puzzle would fall into place.
 Step 1

The first step is to solve the inviscid flow equations (e.g., Euler) for a given airfoil shape including an estimate for the displacement effects \(\delta ^*\). This results in the streamwise flow velocity U, which then drives the boundary layer.
 Step 2

The second step is to solve, for this given U, the two differential equations that describe the boundary layer: von Kármán’s equation (2) and Head’s entrainment definition (4). This provides new values for the momentum thickness \(\theta \) and Head’s shape factor \(H_{\delta \delta ^*}\).
 Step 3
 In the third step, algebraic closure relations are evaluated for the remaining variables:
 (a)

The entrainment E follows from \(H_{\delta \delta ^*}\) using Head’s first assumption (5).
 (b)

The conventional shape factor H follows from \(H_{\delta \delta ^*}\) using Head’s second assumption (6).
 (c)

The displacement thickness \(\delta ^*\) follows from \(\delta ^*= H\,\theta \).
 (d)

The shear stress coefficient \(c_f\) follows from, e.g., a Ludwieg–Tillmann relation [27] or Green’s modification [28] (details turn out not to be essential, therefore we refrain from showing these relations explicitly).
Experimental evidence for this minimum came at the end of the 1970s [29, 30] and in the 1980s [31, 32]; see Fig. 9 (which has been inspired by a graph in [33]). The figure also shows the theoretical models that were proposed for the H–\(H_{\delta \delta ^*}\) relation. The first proposals by Green [34], East [35], and Cebeci and Bradshaw [36] were hesitating to include a minimum in the H–\(H_{\delta \delta ^*}\) relation. Le Balleur at ONERA [26] was the first to explicitly consider such a minimum. He was followed by Houwink and the author at NLR [28] and Lock at RAE [37].
6 Separated flow, at last ...
Around the same time at the Netherlands Aerospace Center NLR in Amsterdam, the author, inspired by the paper by Catherall and Mangler [22] but not yet aware of Le Balleur’s achievements, was studying the relation between U and \(\delta ^*\) of solutions of the full boundarylayer equation (1) [38, 39]. It turned out that this relation shows a similar minimum (Fig. 11) at or close to the point of flow separation. However, this relation is position dependent and far from being universal, unlike Head’s expectation for the H–\(H_{\delta \delta ^*}\) relation.
Thus, since the late 1970s two methods were available to simulate separated boundary layers. Their common essential idea is that the edge velocity U is not prescribed to the boundary layer, but results from the solution of the coupled viscous–inviscid interaction (VII) problem. A comparison between both approaches has been made in the survey paper by Lock and Williams [33].
It is interesting to see what the effect is of the various H–\(H_{\delta \delta ^*}\) closure relations shown in Fig. 9. Hereto calculations have been performed of flow around a NACA 0012 airfoil at a Reynolds number of 9 million at varying angle of attack. The flow was tripped at 0.014 chord at the upper side and at 0.7 chord at the lower side. The turbulent parts of the flow were modeled with the above entrainment model for various H–\(H_{\delta \delta ^*}\) closures. The laminar part was modeled using Falkner–Skan closure. Further details can be found in the PhD theses of Edith Coenen [40] and Henny Bijleveld [41]. The latter thesis also describes the case of unsteady flow and the link between the H–\(H_{\delta \delta ^*}\) closure and the Van Dommelen–Shen [42] singularity, which occurs when \(\mathrm{d} H_{\delta \delta ^*} / \mathrm{d}H = H_{\delta \delta ^*}/H \).
Looking at the slope of the H–\(H_{\delta \delta ^*}\) relation for larger values of H, it is to be expected that Cebeci–Bradshaw’s closure is the most robust, whereas Le Balleur’s closure is the least robust. The most accurate results are to be expected from the closures by Lock and Houwink. This is confirmed by the simulation results. Figure 12 shows the lift polar compared with experimental data [43, App. IV]. The four curves have been continued until the calculations break down. For larger angles of attack eventually all calculations break down because the physics of the flow does not allow a steady solution anymore. Observe that the comparison with experimental data can be pretty good, even beyond maximum lift. Not bad for such simple models...!
6.1 Mathematical basis
Attempts to complete Goldstein’s analysis with a firm mathematical basis have failed thus far; existing mathematical theory is still restricted to accelerating flow [44, Chap. 2]. Yet, the evidence for a relation between Goldstein’s singularity and nonexistence of a solution of the boundarylayer equations at prescribed edge velocity U is substantial. Next to the alreadymentioned conjectured singularity in the \(\delta ^*\)–U relation from the full boundarylayer equations, approximate calculations using (laminar) Falkner–Skan velocity profiles analytically show such a singularity [45, 46]. Also, the strong interaction between boundary layer and local inviscid flow region in the asymptotic tripledeck theory, with its reversal of hierarchy, points in this direction; see, e.g., [1, 2, 23, 47, 48]. The latter asymptotic theory can be extended to describe boundarylayer flow with marginal separation [9, 49], until the separation bubble becomes unsteady and vortexshedding sets in [50] (in terms of Fig. 11, there is no intersection anymore of the inviscid flow relation and the boundarylayer relation).
6.2 Relation with Navier–Stokes
The singularity for prescribed U will not disappear when the full Navier–Stokes equations are used in the boundary layer. Only when a thicker computational domain is used containing part of the adjacent inviscid flow region (which gives the prescribed U the opportunity to adapt itself to the presence of the boundary layer), the singularity can be avoided (unpublished results).
During the 1980s, Navier–Stokes methods had a tough job in keeping up with these VII methods. In the viscous transonic flow workshop [51] in 1987, many methods have been compared for simulating several airfoils under transonic flow conditions. In those days, the stateoftheart in turbulence modeling for the Navier–Stokes methods was not good enough to predict maximum lift (most Navier–Stokes methods did not even produce a maximum). In contrast, the VII integral methods, using the above approach, did a good job.
7 What will the future bring?
In the three decades since the mid1980s, much progress has been made in simulating separated flow using a Navier–Stokes model. This progress has been achieved partly because of improvements in computational hardware. But also improvements in computational algorithms can be mentioned, of which the most important one is the advent of energypreserving discretization methods, e.g., [52, 53, 54]. The current stateoftheart are direct numerical simulations at Reynolds numbers around 100,000. An example is presented in Fig. 13 showing a direct numerical simulation (DNS) of a massively separated flow around a delta wing at Re = 150,000 featuring natural transition to turbulence. The flow has been calculated by Wybe Rozema [55] using a numerical method without artificial diffusion and implemented in the earlier mentioned NLR method Enflow.
For larger Reynolds numbers, in cruise flight in the order of 50 million, DNS will be possible in a few decades from now. Until that time, resort has to be sought to largeeddy turbulence models, again using as little eddy diffusion as possible. A modern example is the anisotropic minimum dissipation model AMD [56, 57].
8 Epilogue
Describing the physics in aerodynamic boundary layers has provided a centurylong modeling inspiration. In particular, the quest for a model that can describe separated boundary layers spanned three quarters of a century. Also, as shortly touched upon, it strongly stimulated the development of singular perturbation theory, e.g., [24, 58, 59].
With hindsight, we can say that Head’s experimentfitted H–\(H_{\delta \delta ^*}\) curve, based on a simple entrainment model, tells it all! Yet, Head may not have foreseen its impact in the mid1950s when he proposed to make this relation central to his boundarylayer model. It was not until two decades later that a minimum in this relation showed up, explaining the poorly understood breakdown of boundarylayer calculations for separated flow. This then opened the way for cheap engineering calculations of aerodynamic boundary layers, which are still competitive for airfoil and wing design with limited separated flow regions (as in cruise flight), e.g., [60, 61, 62].
This leaves us with the question: “Where did Head find the inspiration that made him focus on the H–\(H_{\delta \delta ^*}\) relation?” It was not just ‘big data’ analytics avant la lettre. The story shows that to extract useful information from large amounts of data requires expertise in the domain area, and— to end in the same language as we began—Fingerspitzengefühl (literally: finger tip feeling).
Notes
Acknowledgements
Dr. Wybe Rozema is kindly acknowledged for providing the DNS results of the flow past a delta wing.
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