Particle movement in a boundary layer

To solve for h and \(\theta \) in (2.4), we require a relation between the pressure p and the moving underbody shape F(X, T). Note first that the Y coordinate at an arbitrary point is related to the corresponding coordinate at \(X=1\) on its streamline viaWe will, for the purposes of this study, assume exclusively forward flow such that \(u \ge 0\) everywhere, and hence, every streamline passes through both the pre-Euler region and through the position \(X=1\). From (2.7), we have that \(u_0(Y_0) = u(1,Y_1)\), where \(Y_0\) is the Y-coordinate of a given streamline in the pre-Euler region, and thus, by (2.8), \(\mathrm{{d}}Y_0/\mathrm{{d}}Y_1 = 1.\) Since \(Y_0\) and \(Y_1\) are both zero at the wall, \(Y_0=Y_1\) and thus, the velocity profile at the trailing edge is identical to the incoming velocity profile, \(u(1,Y) = u_0(Y)\). The equivalent relation for an arbitrary point (X, Y) is \(dY/dY_1 = u_0(Y_1)/u(X,Y) = u_0(Y_1)/(u_0(Y_1)^2-2p(X))^{1/2}\). Choosing the streamline along the underside of the body, we obtain the pressure–shape relationCoupling this to the body motion equations allows them to be solved numerically. The results given in Fig. 2 show that the body will clash with (impact on) the wall in finite time (given suitable initial conditions). The asymptotic dependency of h and \(\theta \) on time can be determined by first assuming that \(F(X,T) = O(T_0-T)^N,\ N>0\) as the clash is approached, in the X range \(X = X_0+O(T_0-T)^{N/2}\), with the clash occurring at some \((X_0,T_0)\). From (2.9), \(p \sim F^{-2}\) as \(F\rightarrow 0\), and so \(p = O(T_0-T)^{-2N}\). The body motion equations, thus, yield \(N = 4/5\). This value shows very good agreement with numerical results for the behaviour of h and \(\theta \) in the limit of a clash (see Fig. 2), and the results suggest little dependency on the incoming profile.

At the end of the first stage, the body speed \(h_T =O(T_0-T)^{-1/5}\) tends to infinity, and thus, we anticipate the need for a second time stage. There is now an inner region, given by \(X-X_0 = O(T_0-T)^{2/5}\), and an outer region where \(X-X_0\) is O(1). See Fig. 3. Defining the order-unity time coordinate \(s = (T-T_0)/\tau \) for the unknown time scale \(\tau \), the rescaled variables for the inner region, informed by the results of Sect. 2.1, are \( \tau ^{2/5}\xi = X-X_0, \ \tau ^{4/5}\eta = Y, \ \tau ^{-4/5}{\tilde{u}} = u, \ \tau ^{-2/5}{\tilde{v}} = V, \ \tau ^{-8/5}{\tilde{p}}(\xi ,s) = p, \ \tau ^{4/5}\phi (\xi ,s) = F(X,T), \ \tau ^{4/5}({\tilde{h}},{\tilde{\theta }}) = (h-h_0,\theta -\theta _0). \) Hence, we find the new governing equationsand boundary conditionsThe time derivatives have coefficient \(\varepsilon \tau ^{1/5}\) and so remain small for the current time scale. The horizontal velocity \({\tilde{u}}\) is effectively independent of \(\eta \) because the vorticity \(-u_Y\) is conserved to leading order on streamlines (by the vorticity equation), and so must remain O(1) in the inner region as in the outer region, leading to \({\tilde{u}}_\eta = O(\tau ^{8/5})\). Hence, \({\tilde{u}}\) and \({\tilde{p}}\) are expressible in terms of the volume flux q(s),The underbody curve in the inner region can be approximated as \(\phi (\xi ,s) = \alpha \xi ^2/2+{\tilde{h}}(s)+(X_0-\beta ){\tilde{\theta }}(s), \) where we have defined \(\alpha = F_u''(X_0)\).

In the outer region, the velocity and pressure are of order unity. The \(Vu_Y\) term is not negligible here; so instead we rewrite the X-momentum equation in (2.1) in the form \( {\mathbf {u}}\cdot \nabla (u^2/2+p) = -\varepsilon \tau ^{-1}uu_t \) which becomeswhere we have used \(\mathrm{{d}}s' = \mathrm{{d}}X/u\). Here, \(s'\) parameterises streamlines. If we integrate from the left outer region to the right outer region, i.e. over the inner region, the integral on the RHS (right-hand side) of (2.13) is dominated by this inner contribution, which is of O(1) for \(\tau = \varepsilon ^{5/7}\), thus determining the present time scale.

Then integrating from 0 to some point on the left giveswhere \(u_L\) and \(p_L\) are the u and p values at the chosen point in the left region, with streamlines coming from \(Y = Y_0\) in the pre-Euler region, while integrating from 0 to some point on the right giveswith \(u_R\) and \(p_R\) defined analogously to \(u_L\) and \(p_L\). In particular, \( u(1,Y_1,t)^2/2 = u_0(Y_0)^2/2-\int _{-\infty }^{\infty } {\tilde{u}}_s\ \mathrm{{d}}\xi . \) Recall that from (2.10), \({\tilde{u}}\) is known in terms of \(\phi \) and the volume flux q(s). The system can be closed as follows. Let \(F_E(s)\) be the Y value of the continuation of the body streamline into the pre-Euler region. Then,Using the modified Bernoulli relation (2.15) and integrating \(\mathrm{{d}}Y_1/\mathrm{{d}}Y_0 = u_0(Y_0)/u_1(Y_1)\) from the wall to the body, we haveThe coupled system of (2.17) with (2.16) amounts to an ODE for q for each h and \(\theta \). Then (2.12) yields a pressure-shape relation which, coupled with (2.4), can then be solved numerically (as in the previous section). The results of the numerical integration are shown in Fig. 4. As \(s \rightarrow 0\), \(\int _{-\infty }^{\infty } {\tilde{u}}_s\ \mathrm{{d}}\xi \) cannot become large or else the integral becomes complex (we can rule out large negative values of the integral as this would correspond to small inner velocity but large outer velocity which contradicts volume flux conservation). Assuming \(\phi (\xi ,s) = O(|s|^N_1)\) as \(s \rightarrow 0\) for \(\xi = O(|s|^{N_1/2})\) and setting these large terms to zero at leading order shows that \(q = O(|s|^{N_1/2})\). This leads to the conclusion that \(N_1\) must be 1 by considering the body motion equations. This value is confirmed by numerical results (see Fig. 4). The body now collides with finite impact speed. This is in line with the result of [6] which investigates a similar scenario in the absence of incoming vorticity, and overall the present work indicates that the qualitative behaviour of the body in the event of impact is largely independent of the detailed incoming flow profile, even if certain features depend on the details.

The body length \(L_1\) and the body thickness \(H_1\) are assumed to be in the range \( \mathrm {Re}^{-3/8} \ll L_1 \ll 1,\quad H_1 = O(\mathrm {Re}^{-1/2}L_1^{1/3}),\) corresponding, respectively, to lying between the development and interactive lengths described in the previous paragraph and to being comparable with the representative viscous thickness. The main governing equations of concern [9] then describe the flow solution in the nonlinear wall layer where the expansionapplies in (1.1), leading to the interactive viscous-inviscid system Here, \(X_1\) is scaled on \(L_1\), \(Y_1\) on \(\mathrm {Re}^{-1/2}L_1^{1/3}\) and the unknown pressure P is independent of \(Y_1\). Further, we take the body as non-translating for now and the oncoming velocity profile \(u_0(Y)\) in the boundary layer as having slope \(du_0/dY = \lambda \) at the wall where \(\lambda \) is a given positive constant and again Y is scaled on \(\mathrm {Re}^{-1/2}\). The appropriate boundary conditions (see also Sect. 2) for the interval \(0< X_1 < 1\) are then These represent in turn the no-slip requirement at the wall, the matching requirement with the core region between the wall layer and the body where inviscid properties hold, and the requirements at the trailing edge where \(P, P_{X_1}\) must vanish. The function \(F_u\) is the prescribed shape of the underbody, while \(K(T_1)\) is an unknown mass flux factor to be determined and time (\(T_1\)) dependence is highlighted in readiness for Sect. 3.3, with \(h_1,\ \theta _1\) denoting the lateral displacement and rotation angle, respectively. The factor \(K(T_1)\) is needed because we do not know in advance how much fluid will pass under the body and how much over it. A nonlinear problem similar to that of (3.2)–(3.3) is addressed in [10]. Jumps in the pressure and velocity across the leading edge station similar to those described in the previous section are also involved in the present configuration.

A linearised solution for small \(O({\hat{\varepsilon }})\) values of \(h_1,\ \theta _1,\ F_u\) is now sought, where \({\hat{\varepsilon }}\) denotes the typical size of the initial displacement, rotation and corresponding velocities as well as the underbody shape function. Thus, we have the expressionsproducing on substitution into (3.2)–(3.3) a linear system with K of \(O({\hat{\varepsilon }})\). Proceeding as in [9, 12] we take an \(X_1\)-wise Fourier transform of the linearised version of (3.2a), (3.2b), apply the conditions (3.3a), (3.3b) to the solution and then invert the transform. In consequence, we find for the flat-plate case of zero \(F_u\) that the pressure solution takes the formwhere \(\gamma \) is a known positive constant and \(a_0\) is an unknown function of \(T_1\). The condition (3.3c) has considerable effect here as it requires both \( P_1 = 0\text { and }P_{1X_1} = 0\text { at }X_1 = 1^{-}. \) Hence from (3.5) we obtain the relationsdetermining the relative mass flux in the gap and the amount of upstream influence in terms of the angle \(\theta _1\): a plot of the pressure (3.5) is presented in Fig. 5. It follows also that there is significant growth of the flow perturbations as the leading edge is approached. This growth is provoked by the trailing edge requirements, and it implies there is more subtle interaction close to the leading edge through triple-deck and Euler regions.

The Euler problem takes a linear form, namely The boundary conditions of tangential flow at the solid surfaces, assuming there is no significant separation [5‐14], are where \(w(T_1)\) acts as a given positive constant. The form (3.7a, 3.7b) coupled with appropriate far-field conditions represents a rather basic problem. A conformal mapping \(\pi z/w = - \log (z_2)+z_2/2 + \log (2)-1 + \mathrm{{i}}\pi \) can be used to transform the flow region \(z = x_1+iY\) to an upper half plane \(z_2 = x_2 + iy_2\). For the example of a body relatively close to the wall, such that w is small, the oncoming velocity profile is \(u_0(Y) = \lambda Y\) to leading order and the solution of (3.7a), (3.7b) islocally. Here, (3.8) produces a pressure variation decaying like the inverse of distance from the origin. On the other hand, there is an outer region where the complete profile \(u_0(Y)\) has influence and so may adjust the pressure variation. The far field beyond that needs further consideration.

As stated at the start of this section, similar reasoning holds for flow in a channel of unit width [5‐9, 12], where the expansion parameter is the Reynolds number \(R_2\) based on the channel width, while the non-dimensional body length \(L_2\) and width \(H_2\) satisfy \( R_2^{1/7} \ll L_2 \ll R_2,\quad H_2 = O(R_2^{-1/3}L_2^{1/3}). \) The governing equations in the linearised scenario are identical with those leading to (3.5) and so the behaviour (3.6) holds exactly [9, 12].

Closer to the leading edge, however, the channel flow has a viscous-inviscid interaction over the streamwise length scale \(R_2^{1/7}\), with pressure and velocity jumps across that edge. The smoothing-out of the \(X_1^{-2/3}\) effect appearing in (3.5) occurs over this length scale because of the action of the lateral pressure gradient \(\partial p/\partial y\) as in [12, 34, 35]. Given the eigenfunction behaviour of displacement upstream of the leading edge and the comparative smallness of displacement immediately thereafter, the scaled pressure downstream is given by the formula [12]:where \(\kappa \) is a known positive constant. Specifically \(\kappa = ( 6 \mathrm {{Ai}}^{\prime }(0)/ J)^{3/7} \lambda ^{5/7}\) with J being the integral of \(u_0^2(Y)\) with respect to Y across the channel from 0 to 1 and the positive constant \(\lambda \) being the scaled wall shear stress of the incident velocity profile \(u_0(Y)\); here \(\lambda = 1\) in the case \(u_0(Y) = Y(1 - Y)\) of oncoming Poiseuille flow. The pressure \(P_2\) is finite at \(X_2 = 0^+\) and decays like \(X_2^{-2/3}\) as \(X_2 \rightarrow \infty \) downstream, consistent with (3.5). See Fig. 6. (A similar smoothing-out takes place in the case of the boundary layer.) The earlier-mentioned jumps across the leading edge are smoothed out by an Euler region in which (3.7a), (3.7b) hold along with Here, \(x_1\) is now identical with x, while (3.10b), (3.10c) provide consistency with (3.7b), (3.10a) and boundedness in the far field. Moreover \({\hat{\psi }}, {\hat{p}}\) denote the main perturbations which can be as large as the order \(R_2^{-2/7}\) for the limit case where \(L_2\) tends to \(O(R_2^{ 1/7})\) and \({\hat{\varepsilon }}\) tends nominally to O(1), with \({\hat{p}}\) given by Here, \({\hat{p}} \rightarrow 0\) exponentially in the far field, thus, allowing a match with the pressure over the longer \(R_2^{1/7}\) scale, and \({\hat{p}}\rightarrow 0\) at each of the channel walls, where there is a significant inviscid sublayer. The sublayer width \(O(R_2^{-2/7})\) and length O(1) join the upstream and downstream parts of the viscous-inviscid interactive solution mentioned above. The sublayer velocity profiles far upstream and downstream are different from each other since the flow solution in the Euler region allows a difference in displacement because of (3.10c) but the integral connection between the two profiles is similar to that described in the Sect. 2, and hence, the jumps in pressure and displacement are determined. These jumps match with those of [12] in the linearised case.

A computational solution of the Euler problem (3.7a), (3.7b), (3.10a)–(3.10c) is presented in Fig. 7. This includes plots of the pressures \({\hat{p}}\) acting on the body, with \(p_{\text {ABOVE}},\ p_{\text {BELOW}}\) denoting \({\hat{p}}\) evaluated at Y equal to \(w^+,\ w^-\) in turn. A significant property here is that this pressure \({\hat{p}}\) acting locally to the leading edge gives a contribution to the lift \(C_L\) which can be as large as the order \(R_2^{-2/7}\) for the limit case, thereby outweighing the contribution of order \(R_2^{-2/3}L_2^{5/3}\) (approximately) from the remainder of the body, for a range of body lengths. This is abetted by the moment \(C_M\) likewise being susceptible to domination by \({\hat{p}}\) rather than by the pressure effects from elsewhere on the body surface.

Concerning the evolution of the fluid flow and body motion when the body is free to move, the scaled lift \(C_L\) and moment \(C_M\) clearly play crucial roles because of the relevant body-balance equations which are as in (2.4) essentially.

In channel flow, the dominant contributions to the lift and moment can stem from the vicinity of the leading edge by virtue of the pressure in (3.10d). This reaction is such that when the body inclination \(\theta _1\) is positive, then due to the viscous effect on mass flux, the corresponding \(C_M\) turns out to be positive. Thus, in Fig. 7 where the normalised flux is unity, the pressure perturbation \({\hat{p}}\) is positive on top but negative underneath the body and so yields a positive moment (and negative lift) on the body, an aspect which makes physical sense. Hence, the angular acceleration is positive owing to (2.4). It follows that the fluid–body interaction here is unstable over the present time scale.

In the boundary–layer setting, a similar phenomenon may well arise. This is not fully clear as yet because of the nature of the Euler region and the necessity of matching with the viscous-inviscid solution over the \(\mathrm {Re}^{-3/8}\) length scale upstream and downstream. The flow structures for the boundary-layer and channel-flow cases are fairly similar, but the exponential decay of pressure found in the channel-flow setting is unlikely to apply for the boundary layer, while algebraic decay as mentioned in Sect. 3.1 leads to a divergent integral for the local lift and moment. This in turn suggests that the surface pressure on the remainder of the body plays a pivotal part in the evolution.

Altogether we can expect the overall fluid-body interaction for boundary layers, wall jets and pipe or channel flows to be linearly unstable over the time scale of interest. This makes physical sense in the context of flow at high Reynolds number past a thin body at incidence although we note that the novel time scale is that of the body motion rather than that of a high-frequency instability for instance. The work tends to confirm a trend towards a blockage effect even though viscosity is included; nevertheless stabilising measures are possible as described by [9, 11]. As far as body translation is concerned, we can also expect a match with the properties in Sect. 2 when the present streamwise body speed is increased from zero to a size \(\gg L_1^{1/3}\) (see (3.1)) as discussed in [8, 10] for the boundary-layer setting and likewise for the channel flow.

Eight different scenarios are studied here in which the leading edge gap is varied (\(\beta _1 = 0.1\) and 0.05), the trailing edge gap varied relative to \(\beta _1\) (\(\beta _2 = \beta _1 + 0.1\) and \(\beta _2 = \beta _1 + 0.05\)) and \(u_w\) is either positive or negative (\(u_w = 0.1\) and \(-0.1\)). For each case, the local Reynolds number R is 10.

Velocity profiles across the body are presented in Figs. 9 and 10. In Fig. 9, where \(u_w = 0.1\), for \(\beta _1 = 0.1\) (the first column), there is no flow reversal. However, as \(\beta _1\) decreases, flow reversal is seen in both \(\beta _2\) cases in the gap between the body and the wall, close to the trailing edge. This separation is induced by the closing of the gap between the wall and body which begins to cut off the mass flux beneath the body. For decreased inclination (\(\beta _2 = \beta _1 + 0.05\)), reversal is also seen at a second location close to the leading edge. With the reduced inclination the location of the smallest gap moves towards the centre of the body and, due to the body’s elliptical shape, also decreases in size. The incoming flow is thus squeezed by the diminishing gap and begins to turn when the gap becomes insufficiently wide along the curve of the body, starting to cut off the fluid flux below the body.

For \(u_w = -0.1\), the velocity profiles in Fig. 10 are markedly different. Concerning when flow reversal occurs in these scenarios, note that the free-stream velocity has \(u \le 0\) for \(y \le 0.1\) and \(u > 0\) for \(y > 0.1\). Starting with the top left plot, the leading edge of the body is stationed at the cusp of the change in velocity direction, \(\beta _1 = 0.1\). For \(\beta _2 = \beta _1 + 0.1\), flow separation is seen under the body, close to the trailing edge with the velocity remaining negative for values of \(y > 0.1\). As \(\beta _1\) decreases (top, right), flow reversal of a different nature is seen at several points under the body with the velocity becoming positive for \(y \le 0.1\) and ahead of the body (above the leading edge). Thus, eddies are forming at the leading and trailing edges. A similar trend is seen for the \(\beta _2 = \beta _1 + 0.05\) cases. The notable differences are that slight flow reversal exists ahead of the body even for larger \(\beta _1\) (bottom, left) and that with decreasing \(\beta _1\), there is flow reversal at the leading edge both above and below (with the velocity becoming positive in the gap).

To examine the system further, plots for the wall pressure are presented in Fig. 11 for \(u_w = 0.1\) and Fig. 12 for \(u_w = -0.1\). Qualitatively, the results are remarkably similar (up to a difference in sign), and several features stand out. Concerning flow separation, the magnitude and sign of the pressure gradient are important. In Figs. 11 and 12, there is an initial localised jump in pressure towards the leading edge, producing an adverse pressure gradient (given the velocity direction), which indicates the possibility of flow reversal ahead of the body. After this jump, the pressure gradient becomes favourable, until around the body mid-point where it becomes adverse once again, continuing through to the trailing edge and levelling out as the far field is approached. These results show that as the leading edge of the body becomes closer to the wall (right-hand columns), the amplitudes in pressure are far greater than the equivalent larger gap scenarios (left-hand columns), indicating an increased likelihood of flow reversal with the body’s proximity to the wall. Likewise, in the scenarios with decreased inclination (bottom rows), the magnitudes of the gradients are larger throughout the domain than their increased inclination counterparts; hence, the onset of flow reversal is more likely in both the upstream and downstream regions for the less inclined bodies. These results are consistent with the dynamics and trends previously seen in Figs. 9 and 10 and in the analysis elsewhere in the paper. They also help to establish a trend in the expected dynamics as the gap size and inclination change.

In Sect. 4.1, the effect of distance between the body and the wall has been of chief concern as impact is neared. The increased likelihood of the occurrence of significant flow reversal beneath the body (with decreased wall gap) and ahead of the body (for decreased inclination) is notable. The position and magnitude of the separation are highly dependent on the inclination of the body, with a more parallel formation producing significant reversal ahead of and behind the body. It is expected that the results of [13] concerning the effects (on upstream influence and separation) of different Reynolds numbers and different magnitudes of wall velocities for a flat plate will also hold in this scenario. Agreement with reduced-system analysis for a flat plate is found in [14].

The reduced-system analysis also applies when the ellipse is sufficiently close to the wall for large Reynolds numbers, with the interactive boundary-layer equations (3.2a, 3.2b) again describing the flow around the body. Unsteadiness due to the ellipse motion leads to the response in Fig. 13 obtained from numerical time marching. Flow reversals near the leading and trailing edges appear eventually, with the underbody shear changing sign. The results point to a mid-chord impact different from those observed for a flat body [13, 14]. The latest-time plot in Fig. 13b is tentative because of the severe numerical challenge near termination, which accentuates the desirability of further analysis.