Manipulating flow separation: sensitivity of stagnation points, separatrix angles and recirculation area to steady actuation

Abstract

A variational technique is used to derive analytical expressions for the sensitivity of several geometric indicators of flow separation to steady actuation. Considering the boundary layer flow above a wall-mounted bump, the six following representative quantities are considered: the locations of the separation point and reattachment point connected by the separatrix, the separation angles at these stagnation points, the backflow area and the recirculation area. For each geometric quantity, linear sensitivity analysis allows us to identify regions which are the most sensitive to volume forcing and wall blowing/suction. Validations against full nonlinear Navier−Stokes calculations show excellent agreement for small-amplitude control for all considered indicators. With very resemblant sensitivity maps, the reattachment point, the backflow and recirculation areas are seen to be easily manipulated. By contrast, the upstream separation point and the separatrix angles are seen to remain extremely robust with respect to external steady actuation.

1. Introduction

Flow separation leads in many aerodynamic situations to performance loss, such as reduced lift, increased drag, enhanced fluctuations or noise production. By contrast, separation yields a recirculation region that is often desirable in combustion devices. It is thus not a surprise that there is extensive research on the control of flow separation [1]. Attempts include, in decreasing order of complexity, closed-loop separation control, harmonic or steady active open-loop control and passive control devices.

Studies on closed-loop control strategies remain few: while Alam et al. [2] have provided an analytical approach to closed-loop separation control, based on a kinematic theory of unsteady separation [3], most experimental approaches rely either on low-order reduced models, extracted by physical means or using identification methods [4], or on the design of black box controllers [5,6].

Open-loop control has been successfully applied to separation control: harmonically pulsed synthetic jets [7,8], as well as steady suction or blowing at the wall (e.g. [9] and other references in [10], or [11] as a combination of steady suction and pulsed blowing). The determination of the best placement and frequency of the actuators was often left to extensive parameter sweeps resorting to intensive experimental or computational campaigns.

Passive control strategies rely on the optimization of the geometry or on the addition of appendices, such as vortex generators [12]. Their efficient and robust design requires ideally so-called sensitivity maps. These maps allow one to choose the most sensitive regions and therefore to design optimal control configurations, but again at the expense of intensive experimental or computational campaigns. A recent example is provided by Parezanović & Cadot [13], who studied the influence of a thin control wire on the frequency, drag and recirculation length of the wake behind a D-shaped cylinder.

In all these control methods, an essential aspect is determining meaningful and unambiguous control variables. Among these the manipulation of the separation or reattachment locations [2,14], the recirculation length [15] or the recirculation area [6] appear as valuable and accessible geometric descriptors of the flow. The existence of a link between the geometric properties of separated flows and their stability properties and associated aerodynamic loads is indeed now well accepted.

It is known for instance that the destabilization of the wake of a bluff body beyond the critical Reynolds number takes place simultaneously with a decrease in the recirculation length caused by the mean flow distortion maintained by the progressive development of the instability [16]. Therefore, if one is willing to enhance mixing or reduce flow-induced structural vibrations, then it is natural to target the recirculation length. More recently, Parezanović & Cadot [13] established a clear correlation between base pressure increase (and therefore drag reduction) and mean recirculation area increase in the wake of a D-shaped cylinder at Re∼10⁵, suggesting the direct targeting of the separation properties as a promising control strategy. It is also worth noticing that, in some control attempts [17,18], without being directly targeted, a modification of the recirculation length was observed as a by-product of the control scheme.

This study is dedicated to the determination of analytical expressions for the sensitivity to steady actuation of the following six geometric indicators of flow separation: the locations of the two separation points which connect the separatrix, the separation angles prevailing at these separation points, the backflow area and the recirculation area. These geometric quantities can easily be measured in experiments. Sensitivity maps are computed by solving adjoint equations at the same cost as that of computing the uncontrolled flow, and these maps allow one to identify sensitive regions and predict the effect of small-amplitude control without actually computing any controlled flow. The main focus of this study is on the rigorous derivation of sensitivity expressions for steady control, which constitutes a first step towards more general unsteady control. The flow configuration considered is the boundary layer flow above a wall-mounted bump, on which several open-loop [19] and closed-loop [18] control strategies have been tested numerically.

This paper is organized as follows: geometric quantities of interest are introduced in §2; the concept of sensitivity analysis is presented in §3; expressions of sensitivity to base flow modification are derived in §4; results and validation for volume control and wall control are given in §5.

2. Characteristic quantities in separated flows

The evolution of the recirculation length in separated flows with the increase in Reynolds number Re is well documented [16,20–26]. For instance, in the flow around a circular cylinder, the recirculation length is known to increase with Re in the steady laminar regime, while it starts to decrease in mean value as Re is further increased in the unsteady laminar regime. In this study, we turn our attention to several characteristic quantities which describe the separation, in complement to the recirculation length: the locations of the two separation points (x_s and x_r) which connect the separatrix, the separation angles (α_s and α_r) prevailing at these separation points, the backflow area A_back and the recirculation area A_rec.

As an archetypical flow configuration, we consider the boundary layer flow above a wall-mounted bump studied through direct numerical simulation [24] and through global stability analysis [27].

Figure 1 is a schematic of typical flow separation. The recirculation region is delimited by the wall and the separating streamline. This particular streamline, or separatrix, makes angles α_s and α_r at the separation point x_s and reattachment point x_r, respectively. The wall geometry is described by y_w(x) and the separatrix by y_sep(x). Unit vectors tangent and normal to the wall are noted t and n (figure 2), and ∂_t and ∂_n stand for derivatives along t and n.

Figure 1.

Sketch of the bump flow. The flow separates at x_s and reattaches at x_r (black circles). The separatrix (blue solid line) encloses the recirculation region (blue shading) of area A_rec and makes angles α_s, α_r with the wall. Velocity profiles are shown with black lines. The curve where U_x=0 (red dashed line) delimits the backflow area (red shading) of area A_back. The bump wall geometry is parametrized by y_w(x) and the separatrix by y_sep(x). The axes are not to scale.

Figure 2.

Sketch of tangent and normal vectors to the wall at (x,y)=(x,y_w(x)). (Online version in colour.)

In this paper, we focus on the following quantities.

• (i) The location of stagnation points, i.e. separation point x_s and reattachment point x_r, characterized by zero wall shear stress

  $${\tau }_{\mathrm{s}/\mathrm{r}}={\mathrm{\partial }}_{\mathrm{n}}{U}_{\mathrm{t}}({\mathbf{\text{x}}}_{\mathrm{s}/\mathrm{r}})=0.$$ 2.1
• (ii) The angle between the separatrix and the wall at the separation and reattachment points, given by Lighthill's formula [28]

  $$\tan ({\alpha }_{\mathrm{s}/\mathrm{r}})={-3\frac{{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}}{{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}}|}_{{\mathbf{\text{x}}}_{\mathrm{s}/\mathrm{r}}}.$$ 2.2
• (iii) The area of the backflow region

  $$A_{\mathrm{back}}={\iint }_{\mathit{\Omega }}{1}_{{\mathit{\Omega }}_{\mathrm{back}}}(\mathbf{\text{x}})\hspace{0.17em}\mathrm{d}\mathit{\Omega },$$ 2.3

  where ${\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{back}}}$ is the characteristic function of Ω_back={(x,y) | U_x<0}.
• (iv) The area of the recirculation region enclosed between the separatrix and the wall

  $$A_{\mathrm{rec}}={\iint }_{\mathit{\Omega }}{\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{rec}}}(\mathbf{x})\hspace{0.17em}\mathrm{d}\mathit{\Omega },$$ 2.4

  where ${\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{rec}}}$ is the characteristic function of Ω_rec={(x,y) | x_s≤x≤x_r,y_w(x)≤y≤y_sep(x)}.

The steady-state flow U(x) is calculated by solving the Navier−Stokes equations with a finite-element method and an iterative Newton procedure on a mesh highly refined near stagnation points. A two-dimensional triangulation of the computational domain is generated with the finite-element software FreeFem++ (http://www.freefem.org), and equations are solved in their variational formulation, with the following boundary conditions: Blasius profile at the inlet, no-slip condition on the wall, symmetry condition at the top border and convective condition −Pn+Re⁻¹∇Un=0 at the outlet. P2 and P1 Taylor–Hood elements are used for spatial discretization of velocity and pressure, respectively (see also details and validation in [15,19]). Stagnation points are found according to (2.1) with a bisection on the wall shear stress ∂_nU_t. Angles are calculated using Lighthill's formula (2.2) and are found to agree very well with geometric angles measured between the wall and the separatrix integrated from U(x) with a fourth-order Runge–Kutta method. Areas (2.3) and (2.4) are computed with a trapezoidal rule for the two-dimensional integration of the backflow region and recirculation region.

Figure 3 illustrates how the backflow and recirculation regions grow with Reynolds number in the bump flow. Figure 4 shows that the reattachment point moves downstream linearly with Re, which is typical of steady separated flows. The areas A_back and A_rec show the same trend but increase more than linearly because the backflow and recirculation regions become not only much longer but also slightly higher. The separation point moves a little upstream but stays downstream of the bump summit (x_b=25). The reattachment angle is fairly constant, 180−α_r≃13−15°. The separation angle measured relative to the wall decreases from α_s=19° to 13° between Re=100 and 700, but measured relative to e_x it is small and almost constant, α_s+θ_wall≃6−8°. It is interesting to note that the separatrix angles vary slowly with Re even though the recirculation length increases significantly. These observations indicate that the main effect of Re is to elongate the recirculation region, while the flow remains mostly horizontal.

Figure 3.

Separatrix (blue solid line) and curve of zero streamwise velocity U_x=0 (red dashed line), at Re=100, 300, 500 and 700. The axes are not to scale.

Figure 4.

Variation with Reynolds number of characteristic separation quantities: recirculation and backflow areas A_rec and A_back, separation and reattachment locations x_s and x_r, separation and reattachment angles α_s and α_r relative to the wall (the dashed line shows the ‘absolute’ separation angle α_s+θ_wall, i.e. measured relative to e_x). (Online version in colour.)

3. Sensitivity analysis

In this section, analytical expressions are given for the sensitivity of quantities of interest (2.1)–(2.4) to flow modifications, volume forcing and blowing/suction at the wall.

The sensitivity to flow modification of a quantity of interest, say ϕ, is a field defined through the first-order variation δϕ induced by a small flow modification δU,

$$\delta \varphi =({\mathbf{\nabla }}_{\mathbf{U}}\varphi \hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U}),$$ 3.1

and it can be computed as follows:

$$\frac{\mathrm{d}\varphi }{\mathrm{d}\mathbf{U}}\mathit{\delta }\mathbf{U}=\underset{ϵ\to 0}{lim}\frac{\varphi (\mathbf{U}+ϵ\mathit{\delta }\mathbf{U})-\varphi (\mathbf{U})}{ϵ}.$$ 3.2

Here, $(\mathbf{a}\hspace{0.17em}|\hspace{0.17em}\mathbf{b})={\iint }_{\mathit{\Omega }}\mathbf{a}\cdot \mathbf{b}\hspace{0.17em}\mathrm{d}\mathit{\Omega }$ denotes the two-dimensional inner product in Ω. In other words, the sensitivity ∇_Uϕ=dϕ/dU is the gradient of ϕ with respect to flow modification. Analytical expressions for the sensitivity to flow modification of the geometric quantities of interest considered in this study will be derived in §4 (see (4.10) for the location of stagnation points, (4.24) for separatrix angles and (4.36)–(4.42) for backflow area and recirculation area).

In practice, the base flow cannot be modified arbitrarily and one has to resort to an external control (passive obstacle, heating, magnetic field, geometry modification, wall motion, wall blowing or suction, etc.). This control in turn alters the velocity field. Here, we focus on steady control, either in the domain Ω by means of a body force (source of momentum) C or at the wall Γ_w by means of blowing/suction with velocity U_c. However, the method is general and easily handles other types of control.

Sensitivities to volume control and wall control can be defined through the variation δϕ induced by small-amplitude control,

$$\delta \varphi =({\mathbf{\nabla }}_{\mathbf{C}}\varphi \hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{C})+⟨{\mathbf{\nabla }}_{{\mathbf{U}}_c}\varphi \hspace{0.17em}|\hspace{0.17em}\mathit{\delta }{\mathbf{U}}_c⟩,$$ 3.3

where $⟨\mathbf{a}\hspace{0.17em}|\hspace{0.17em}\mathbf{b}⟩={\int }_{{\mathit{\Gamma }}_{\mathrm{w}}}\mathbf{a}\cdot \mathbf{b}\hspace{0.17em}\mathrm{d}\mathit{\Gamma }$ denotes the one-dimensional inner product on Γ_w. Sensitivities ∇_Cϕ=dϕ/dC and ∇_Ucϕ=dϕ/dU_c can be computed as follows:

$$\frac{\mathrm{d}\varphi }{\mathrm{d}\mathbf{C}}\mathit{\delta }\mathbf{C}=\underset{ϵ\to 0}{lim}\frac{\varphi (\mathbf{C}+ϵ\mathit{\delta }\mathbf{C})-\varphi (\mathbf{C})}{ϵ}\mathrm{and}\frac{\mathrm{d}\varphi }{\mathrm{d}{\mathbf{U}}_c}\mathit{\delta }{\mathbf{U}}_c=\underset{ϵ\to 0}{lim}\frac{\varphi ({\mathbf{U}}_c+ϵ\mathit{\delta }{\mathbf{U}}_c)-\varphi ({\mathbf{U}}_c)}{ϵ}.$$ 3.4

Taking into account the definition of ϕ, and enforcing the Navier−Stokes equations to be satisfied by the flow, a Lagrangian method yields the sensitivities

$$\begin{array}{r}{\mathbf{\nabla }}_{\mathbf{C}}\varphi ={\mathbf{U}}^{†}\mathrm{and}{\mathbf{\nabla }}_{{\mathbf{U}}_c}\varphi =-P^{†}\mathbf{n}-{\mathit{Re}}^{-1}\mathbf{\nabla }{\mathbf{U}}^{†}\mathbf{n},\end{array}$$ 3.5

where the adjoint flow (U^†,P^†)^T is the solution of the non-homogeneous linear equations

$$\mathbf{\nabla }\cdot {\mathbf{U}}^{†}=0\mathrm{and}-\mathbf{U}\cdot \mathbf{\nabla }{\mathbf{U}}^{†}+{\mathbf{U}}^{†}\cdot \mathbf{\nabla }{\mathbf{U}}^{\mathrm{T}}-\mathbf{\nabla }{\mathit{P}}^{†}-{\mathit{Re}}^{-1}{\mathbf{\nabla }}^2{\mathbf{U}}^{†}={\mathbf{\nabla }}_{\mathbf{U}}\varphi ,$$ 3.6

with boundary condition U^†=0 at the wall. The forcing term in (3.6) is the sensitivity of ϕ to flow modification defined in (3.1) and (3.2), which must therefore be computed beforehand, using expressions derived in detail in §4 for all quantities of interest (2.1)–(2.4). Using the same finite-element method as for the determination of the base flow, sensitivities (3.5) are obtained by solving the adjoint equations (3.6) in weak form, particularly convenient to express the forcing term ∇_Uϕ.

4. Derivation of sensitivities to flow modification

In this section, analytical expressions are derived for the sensitivity of quantities of interest (2.1)–(2.4) to flow modification. Recall that the wall Γ_w is parametrized by (x,y)=(x,y_w(x)), as shown in figure 2.

(a). Stagnation points

As expressed in (2.1), steady separation and reattachment points x_s=(x_s,y_w(x_s)) and x_r=(x_r,y_w(x_r)) are characterized by zero wall shear stress. Following [15], stagnation points are redefined in terms of characteristic functions

$$x_{\mathrm{s}}={\int }_0^{x^{\ast }}H(\tau (x))\hspace{0.17em}\mathrm{d}x={\int }_0^{x^{\ast }}G(x)\hspace{0.17em}\mathrm{d}x$$ 4.1

and

$$x_{\mathrm{r}}={\int }_{x^{\ast }}^{\mathrm{\infty }}H(-\tau (x))\hspace{0.17em}\mathrm{d}x+x^{\ast }={\int }_{x^{\ast }}^{\mathrm{\infty }}1-G(x)\hspace{0.17em}\mathrm{d}x+x^{\ast },$$ 4.2

where τ(x)=∂_nU_t|_(x,yw(x)) is the wall shear stress, H is the Heaviside step function defined as H(γ<0)=0 and H(γ>0)=1, and x* is any streamwise location inside the recirculation region. As illustrated in figure 5, the integrand in (4.1) is equal to 1 upstream of the separation point, therefore integrating over 0≤x≤x* does indeed yield the coordinate x_s. Similarly, the integrand in (4.2) is equal to 1 upstream of the reattachment point, and integrating in x yields the coordinate x_r.

Figure 5.

Wall shear stress and associated Heaviside functions appearing in the expression of the stagnation points (4.1) and (4.2). (Online version in colour.)

We assume for the sake of clarity that reattachment occurs far enough downstream where the wall is horizontal and τ(x_r)=∂_yU_x|_(xr,yw(xr)), which is verified in practice for all the Reynolds numbers considered. A flow modification δU makes the reattachment point move by the following amount:

$$\begin{array}{rl}\delta x_{\mathrm{r}}& =\underset{ϵ\to 0}{lim}\frac{x_{\mathrm{r}}(\mathbf{U}+ϵ\mathit{\delta }\mathbf{U})-{\mathbf{x}}_{\mathrm{r}}(\mathbf{U})}{ϵ}\end{array}$$ 4.3

$$\begin{array}{r}=\underset{ϵ\to 0}{lim}\frac{1}{ϵ}\int [H(-\tau (x)-ϵ\delta \tau (x))-H(-\tau (x))]\hspace{0.17em}\mathrm{d}x\end{array}$$ 4.4

$$\begin{array}{r}=\int -{\frac{\mathrm{d}H}{\mathrm{d}\gamma }|}_{\gamma =-\tau (x)}\delta \tau (x)\hspace{0.17em}\mathrm{d}x\end{array}$$ 4.5

$$\begin{array}{r}=\int -{(\frac{\mathrm{d}\tau }{\mathrm{d}x}(x))}^{-1}\frac{\mathrm{d}G}{\mathrm{d}x}(x)\delta \tau (x)\hspace{0.17em}\mathrm{d}x\end{array}$$ 4.6

$$\begin{array}{r}=\int -{(\frac{\mathrm{d}\tau }{\mathrm{d}x}(x))}^{-1}\delta (x-x_{\mathrm{r}})\delta \tau (x)\hspace{0.17em}\mathrm{d}x\end{array}$$ 4.7

$$\begin{array}{r}=-\frac{\delta \tau (x_r)}{{\mathrm{d}}_x\tau {|}_{(x_{\mathrm{r}},y_{\mathrm{w}}(x_{\mathrm{r}})}},\end{array}$$ 4.8

where (4.6) comes from the chain rule differentiation (d(1−G))/dx(x)=−dH/dγ|_γ=−τ(x)(dτ/dx)(x), and (4.7) is the result of (dG/dx)(x)=δ(x−x_r) with δ(x) the Dirac delta function, as G increases from 0 to 1 at x=x_r. Finally

$$\delta x_{\mathrm{r}}=({\mathbf{\nabla }}_{\mathbf{U}}{x}_{\mathrm{r}}\hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U})=-{\frac{{\mathrm{\partial }}_y\delta {\mathit{U}}_x}{{\mathrm{\partial }}_{xy}{\mathit{U}}_x}|}_{{\mathrm{x}}_{\mathrm{r}}}.$$ 4.9

The variation of the separation point is obtained in a similar way, with only slight sign differences. First, the chain rule derivation of G(x)=H(τ(x)) reads (dG/dx)(x)=dH/dγ|_γ=τ(x)(dτ/dx)(x). Second, the expression in terms of the Dirac delta is (dG/dx)(x)=−δ(x−x_s) because G decreases from 1 to 0 at x=x_s. Taking into account the wall geometry, one obtains

$$\delta x_{\mathrm{s}}=({\mathbf{\nabla }}_{\mathbf{U}}{x}_{\mathrm{s}}\hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U})={-\frac{1}{\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}}\frac{{\mathrm{\partial }}_{\mathrm{n}}\delta {\mathit{U}}_{\mathrm{t}}}{{\mathrm{\partial }}_{\mathbf{n}\mathbf{t}}{\mathit{U}}_{\mathrm{t}}}|}_{{\mathrm{x}}_{\mathrm{s}}},$$ 4.10

where y_w′=dy_w/dx. Expression (4.10) is valid for the reattachment point too, where y_w′=0, t≡e_x and n≡e_y.

Figure 6 shows the sensitivity of stagnation points to volume control in the streamwise direction C_x=C⋅e_x obtained at Re=100 and 500 using (4.9) and (4.10) and the method presented in §3. Red (resp. blue) regions indicate where a localized, small-amplitude body force oriented along e_x would move stagnation points upstream, δx_s/r>0 (resp. downstream, δx_s/r<0). The separation point is mostly sensitive near x_s. The reattachment point is sensitive near x_r, but also at the bump summit, in the whole shear layer along the separatrix, and in the recirculation region.

Figure 6.

Sensitivity of stagnation points x_s, x_r, with respect to streamwise volume control C_x. (a) Re=100 and (b) Re=500. The blue solid line is the separatrix, and the red dashed line is the curve where U_x=0.

(b). Separation and reattachment angles

It is remarkable but not well known that the angle between the separatrix and the wall can be expressed analytically as a function of flow quantities at the stagnation point, as expressed in (2.2). We recall briefly Lighthill's original presentation [28]. For the sake of simplicity, we assume first that the wall is flat and horizontal, y_w(x)=0. A Taylor expansion of the streamwise velocity near the wall, y≪1, reads

$$U_x(x,y)=U_x(x,0)+{\mathrm{\partial }}_y{U}_x(x,0)y+{\mathrm{\partial }}_{yy}{U}_x(x,0)\frac{y^2}{2}+\mathcal{O}(y^3).$$ 4.11

This expression is conveniently recast using (i) the no-slip condition U_x(x,0)=0, (ii) the vorticity ω(x,0)=−∂_yU_x(x,0), and (iii) the streamwise momentum equation ∂_yyU_x(x,0)=Re ∂_xp(x,0)

$$U_x(x,y)=-\omega (x,0)y+\frac{\mathit{R}\mathit{e}}{2}{\mathrm{\partial }}_{x}p(x,0)y^2+\mathcal{O}(y^3).$$ 4.12

Equivalently, the stream function defined as U_x=∂_yψ reads

$$\psi (x,y)=-\frac{1}{2}\omega (x,0)y^2+\frac{\mathit{R}\mathit{e}}{6}{\mathrm{\partial }}_{x}p(x,0)y^3+\mathcal{O}(y^4).$$ 4.13

The separatrix ψ=0 is thus described by y_sep(x)=3ω(x,0)/Re ∂_xp(x,0) and separates from or reattaches to the wall with the angle α such that

$$\begin{array}{rl}\tan (\alpha )& =\frac{\mathrm{d}{y}_{\mathrm{sep}}}{\mathrm{d}x}=\frac{3}{\mathit{R}\mathit{e}}(\frac{{\mathrm{\partial }}_x\omega }{{\mathrm{\partial }}_{x}p}-\omega \frac{{\mathrm{\partial }}_{xx}p}{{({\mathrm{\partial }}_{x}p)}^2})\\ & =\frac{3}{\mathit{R}\mathit{e}}\frac{{\mathrm{\partial }}_x\omega }{{\mathrm{\partial }}_{x}p}=-3\frac{{\mathrm{\partial }}_{xy}{U}_x}{{\mathrm{\partial }}_{yy}{U}_x}\end{array}$$ 4.14

because ω(x_s/r,0)=0. Equation (4.14) is valid for a curved or inclined wall too, hence recovering (2.2). A longer but similar derivation is possible following the steps of Haller [3] for unsteady flows, and of course the same expression is also obtained if taking all quantities as steady in his final expression.

To derive the sensitivity of the angle, we introduce the function f defined on the wall

$$f(\mathbf{U},x)={\tan }^{-1}(-3{\frac{{\mathrm{\partial }}_{nt}{\mathbf{U}}_{\mathrm{t}}}{{\mathrm{\partial }}_{nn}{\mathbf{U}}_{\mathrm{t}}}|}_{(x,y_{\mathrm{w}}(x))}),$$ 4.15

where U and x are treated as independent variables. The separation and reattachment angles are equal to α_s/r=f(U,x_s/r). Their variation with flow modification is

$$\delta {\alpha }_{\mathrm{s}/\mathrm{r}}=({\mathbf{\nabla }}_{\mathbf{U}}{\alpha }_{\mathrm{s}/\mathrm{r}}\hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U})=(\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{U}}+{\frac{\mathrm{\partial }f}{\mathrm{\partial }x}|}_{x_{\mathrm{s}/\mathrm{r}}}\frac{\mathrm{d}{x}_{\mathrm{s}/\mathrm{r}}}{\mathrm{d}\mathbf{U}}|\mathit{\delta }\mathbf{U}),$$ 4.16

where the first term of the sensitivity is the direct angle variation due to the change in tangential velocity U_t, while the second term is the indirect angle variation due to the displacement of the stagnation points x_s/r. Before deriving in detail each of the terms of (4.16), we give their expression below

$$\begin{array}{r}\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{U}}|\mathit{\delta }\mathbf{U}\right)={-3\frac{A{\mathrm{\partial }}_{nt}\delta U_{\mathrm{t}}-B{\mathrm{\partial }}_{nn}\delta U_{\mathrm{t}}}{A^2+9B^2}|}_{x_{\mathrm{s}/\mathrm{r}}},\end{array}$$ 4.17

$$\begin{array}{r}{\frac{\mathrm{\partial }f}{\mathrm{\partial }x}|}_{x_{\mathrm{s}/\mathrm{r}}}\delta x_{\mathrm{s}/\mathrm{r}}=-3\frac{B^{\prime }A-A^{\prime }B}{A^2+9B^2}\delta x_{\mathrm{s}/\mathrm{r}}\end{array}$$ 4.18

$$\begin{array}{r}\text{and}\delta x_{\mathrm{s}/\mathrm{r}}=\left(\frac{\mathrm{d}{x}_{\mathrm{s}/\mathrm{r}}}{\mathrm{d}\mathbf{U}}|\mathit{\delta }\mathbf{U}\right)=({\mathbf{\nabla }}_{\mathbf{U}}{x}_{\mathrm{s}/\mathrm{r}}\hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U}),\end{array}$$ 4.19

where

$$\begin{array}{r}A(x)={\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}(x,y_{\mathrm{w}}(x)),\end{array}$$ 4.20

$$\begin{array}{r}B(x)={\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}(x,y_{\mathrm{w}}(x)),\end{array}$$ 4.21

$$\begin{array}{r}{A}^{\prime }(x)=3\frac{y_{\mathrm{w}}^{″}}{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{nn}{U}_n+\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{nnt}{U}_{\mathrm{t}}\end{array}$$ 4.22

$$\begin{array}{r}\text{and}{B}^{\prime }(x)=\frac{y_{\mathrm{w}}^{″}}{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}({\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}+2{\mathrm{\partial }}_{nt}{U}_n)-\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{nnt}{U}_n.\end{array}$$ 4.23

Combining equations (4.16)–(4.23) yields

$$\begin{array}{rl}\delta {\alpha }_{\mathrm{s}/\mathrm{r}}& =({\mathbf{\nabla }}_{\mathbf{U}}{\alpha }_{\mathrm{s}/\mathrm{r}}\hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U})\\ & =-\frac{3}{A^2+9B^2}{(A{\mathrm{\partial }}_{nt}-B{\mathrm{\partial }}_{nn}+\frac{A^{\prime }B-B^{\prime }A}{B\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}}{\mathrm{\partial }}_{\mathrm{n}})\delta U_{\mathrm{t}}|}_{x_{\mathrm{s}/\mathrm{r}}}.\end{array}$$ 4.24

Figure 7 shows the sensitivity of separatrix angles to streamwise volume control obtained at Re=100 and 500 using (4.24). Like stagnation points, the separation angle is mostly sensitive near x_s, and the reattachment angle near x_r. These sensitivity maps show complex structures, with regions of opposite signs close to each other. This indicates that a small displacement of the forcing location could result in changing the sign of the variation δα.

Figure 7.

Sensitivity of separation and reattachment angles α_s, α_r, with respect to streamwise volume control C_x. (a) Re=100 and (b) Re=500. The blue solid line is the separatrix, and the red dashed line is the curve where U_x=0.

We now turn to the derivation of the three terms in (4.16). The latter term (4.19) is precisely the variation of stagnation points (4.9) and (4.10). Next, the variation of f with flow modification (at fixed x) is

$$\delta f=\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{U}}|\delta \mathbf{U}\right)=\underset{ϵ\to 0}{lim}\frac{f(\mathbf{U}+ϵ\delta \mathbf{U},x_{\mathrm{s}/\mathrm{r}})-f(\mathbf{U},x_{\mathrm{s}/\mathrm{r}})}{ϵ},$$ 4.25

where, at first order,

$$\begin{array}{rl}& f(\mathbf{U}+ϵ\delta \mathbf{U},x_{\mathrm{s}/\mathrm{r}})={\tan }^{-1}(-3\frac{{\mathrm{\partial }}_{nt}({\mathbf{U}}_{\mathrm{t}}+ϵ\delta {\mathbf{U}}_{\mathrm{t}})}{{\mathrm{\partial }}_{nn}({\mathbf{U}}_{\mathrm{t}}+ϵ\delta {\mathbf{U}}_{\mathrm{t}})})\\ & ={\tan }^{-1}(-3\frac{{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}}{{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}}-3ϵ\frac{{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}{\mathrm{\partial }}_{nt}\delta U_{\mathrm{t}}-{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}{\mathrm{\partial }}_{nn}\delta U_{\mathrm{t}}}{{({\mathrm{\partial }}_{nn}{U}_{\mathrm{t}})}^2})\\ & ={\tan }^{-1}(-3\frac{{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}}{{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}})-\frac{3ϵ}{1+{(-3({\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}/{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}))}^2}\frac{{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}{\mathrm{\partial }}_{nt}\delta U_{\mathrm{t}}-{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}{\mathrm{\partial }}_{nn}\delta U_{\mathrm{t}}}{{({\mathrm{\partial }}_{nn}{U}_{\mathrm{t}})}^2}\\ & ={\tan }^{-1}(-3\frac{{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}}{{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}})-3ϵ\frac{{\mathrm{\partial }}_{nn}{U}_{\mathrm{t}}{\mathrm{\partial }}_{nt}\delta U_{\mathrm{t}}-{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}{\mathrm{\partial }}_{nn}\delta U_{\mathrm{t}}}{{({\mathrm{\partial }}_{nn}{U}_{\mathrm{t}})}^2+9{({\mathrm{\partial }}_{nt}{U}_{\mathrm{t}})}^2},\end{array}$$ 4.26

which yields expression (4.17).

Finally, the variation of $f(\mathbf{U},x)={\tan }^{-1}(-3\mathbf{B}(x)/\mathbf{A}(x))$ with x (for fixed flow conditions) is derived in a similar way, with straightforward composition of derivatives of ${\tan }^{-1}$ and of a quotient

$$\begin{array}{rl}\delta f={\frac{\mathrm{\partial }f}{\mathrm{\partial }x}|}_{x_{\mathrm{s}/\mathrm{r}}}\delta x_{\mathrm{s}/\mathrm{r}}& =-3\frac{1}{1+{(3B/A)}^2}\frac{A{B}^{\prime }-B{A}^{\prime }}{A^2}\end{array}$$ 4.27

$$\begin{array}{r}=-3\frac{B^{\prime }A-A^{\prime }B}{A^2+9B^2}\delta x_{\mathrm{s}/\mathrm{r}},\end{array}$$ 4.28

which yields expression (4.18). However, some care is needed when computing the derivatives of A and B. Although the streamwise derivative of the total velocity field at the wall ∂_xU is related in a simple way to the tangential derivative ∂_tU by geometric considerations,

$${\mathrm{d}}_x\mathbf{U}{|}_{(x,y_{\mathrm{w}}(x))}=({\mathrm{\partial }}_x+y_{\mathrm{w}}^{\prime }{\mathrm{\partial }}_y)\mathbf{U}=\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{\mathrm{t}}\mathbf{U},$$ 4.29

this is true neither for individual velocity components nor for velocity derivatives. For example,

$${\mathrm{d}}_x({\mathrm{\partial }}_n{U}_{\mathrm{t}}){|}_{(x,y_{\mathrm{w}}(x))}\ne ({\mathrm{\partial }}_x+y_{\mathrm{w}}^{\prime }{\mathrm{\partial }}_y)({\mathrm{\partial }}_n{U}_{\mathrm{t}})=\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}\hspace{0.17em}{\mathrm{\partial }}_{\mathrm{t}}({\mathrm{\partial }}_n{U}_{\mathrm{t}}).$$ 4.30

This is because the tangential velocity U_t=U⋅t depends on x not only through U but also through the local tangent vector t=t(x)=t(x,y_w(x)). Similarly, the normal derivative ∂_n=∇⋅n depends on x because the normal vector n=n(x)=n(x,y_w(x)) does. In (4.30), one must therefore take into account d_xt and d_x∂_n. The calculation is straightforward but tedious when expressing all quantities in the basis (e_x,e_y); instead, one can make a systematic use of the nabla operator

$$\begin{array}{rl}{\mathrm{d}}_x({\mathrm{\partial }}_{\mathrm{n}}{U}_{\mathrm{t}})& ={\mathrm{d}}_x((\mathbf{\nabla }\cdot n)(\mathbf{U}\cdot t))\\ & =\mathbf{\nabla }\cdot ({\mathrm{d}}_{x}n)(\mathbf{U}\cdot t)+(\mathbf{\nabla }\cdot n)({\mathrm{d}}_x\mathbf{U}\cdot t)+(\mathbf{\nabla }\cdot n)(\mathbf{U}\cdot {\mathrm{d}}_{x}t)\\ & =\mathbf{\nabla }\cdot (-\frac{y_{\mathrm{w}}^{″}}{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}t)(\mathbf{U}\cdot t)+(\mathbf{\nabla }\cdot n)(\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{\mathrm{t}}\mathbf{U}\cdot t)\\ & +(\mathbf{\nabla }\cdot n)(\mathbf{U}\cdot \frac{y_{\mathrm{w}}^{″}}{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}n)\\ & =-\frac{y_{\mathrm{w}}^{″}}{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{\mathrm{t}}{U}_{\mathrm{t}}+\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}+\frac{y_{\mathrm{w}}^{″}}{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{\mathrm{n}}{U}_n\\ & =\frac{y_{\mathrm{w}}^{″}}{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}({\mathrm{\partial }}_{\mathrm{n}}{U}_n-{\mathrm{\partial }}_{\mathrm{t}}{U}_{\mathrm{t}})+\sqrt{1+y_{\mathrm{w}}^{\mathrm{\prime }2}}{\mathrm{\partial }}_{nt}{U}_{\mathrm{t}}.\end{array}$$ 4.31

Compared with (4.30), two additional terms coming from the derivatives of ∂_n and t clearly appear. The calculation of A′ and B′ follows similar steps.

(c). Backflow area

The backflow area (2.3) can be expressed as

$$A_{\mathrm{back}}={\iint }_{\mathit{\Omega }}{\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{back}}}(x)\hspace{0.17em}\mathrm{d}\mathit{\Omega }={\iint }_{\mathit{\Omega }}\mathbf{H}(-{\mathbf{U}}_x(x))\hspace{0.17em}\mathrm{d}\mathit{\Omega }.$$ 4.32

Its sensitivity is derived in the same vein as that of stagnation points (§4a)

$$\begin{array}{rl}\delta A_{\mathrm{back}}& =\underset{ϵ\to 0}{lim}\frac{A_{\mathrm{back}}(\mathbf{U}+ϵ\mathit{\delta }\mathbf{U})-{\mathbf{A}}_{\mathrm{back}}(\mathbf{U})}{ϵ}\\ & =\underset{ϵ\to 0}{lim}\frac{1}{ϵ}{\iint }_{\mathit{\Omega }}[H(-U_x(x)-ϵ\delta {\mathbf{U}}_x(x))-\mathbf{H}(-{\mathbf{U}}_x(x))]\hspace{0.17em}\mathrm{d}\mathit{\Omega }\\ & ={\iint }_{\mathit{\Omega }}-{\frac{\mathrm{d}H}{\mathrm{d}u}|}_{u=-U_x}\delta U_x(x)\hspace{0.17em}\mathrm{d}\mathit{\Omega }\\ & ={\iint }_{\mathit{\Omega }}{({\mathrm{\partial }}_{\mathrm{n}}{U}_x)}^{-1}(\mathbf{\nabla }{\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{back}}}(x)\cdot n)\delta {\mathbf{U}}_x(x)\hspace{0.17em}\mathrm{d}\mathit{\Omega }\\ & ={\iint }_{\mathit{\Omega }}-{({\mathrm{\partial }}_{\mathrm{n}}{U}_x)}^{-1}{\delta }_{{\mathit{\Gamma }}_{\mathrm{back}}}(x)\delta {\mathbf{U}}_x(x)\hspace{0.17em}\mathrm{d}\mathit{\Omega }\\ & =-{\oint }_{{\mathit{\Gamma }}_{\mathrm{back}}}\frac{\delta U_x}{{\mathrm{\partial }}_{\mathrm{n}}{U}_x}\mathrm{d}\mathit{\Gamma },\end{array}$$ 4.33

where Γ_back is the boundary of the backflow region Ω_back, and ∂_nU_x is the outward derivative normal to Γ_back. Here, we used the chain-rule derivation $\mathbf{\nabla }{\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{back}}}(x)\cdot n=-\mathrm{d}\mathbf{H}/\mathrm{d}u{|}_{u=-{\mathbf{U}}_x}{\mathrm{\partial }}_{\mathrm{n}}{\mathbf{U}}_x(x)$, and a higher order generalization of the one-dimensional relation

$$\int \varphi (x)\frac{\mathrm{d}H(x-x_0)}{\mathrm{d}x}\hspace{0.17em}\mathrm{d}x=\int \varphi (x)\delta (x-x_0)\hspace{0.17em}\mathrm{d}x=\varphi (x_0),$$ 4.34

namely

$$-{\iint }_{\mathit{\Omega }}\varphi (x)\mathbf{\nabla }{\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{back}}}(x)\cdot n\hspace{0.17em}\mathrm{d}\mathit{\Omega }={\iint }_{\mathit{\Omega }}\varphi (x){\delta }_{{\mathit{\Gamma }}_{\mathrm{back}}}(x)\hspace{0.17em}\mathrm{d}\mathit{\Omega }={\oint }_{{\mathit{\Gamma }}_{\mathrm{back}}}\varphi (x),$$ 4.35

where δ_Γback is the two-dimensional delta function associated with ${\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{back}}}$, and n is the outward normal of Γ_back. Therefore,

$$\delta A_{\mathrm{back}}=({\mathbf{\nabla }}_{\mathbf{U}}{A}_{\mathrm{back}}\hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U})=-{\oint }_{{\mathit{\Gamma }}_{\mathrm{back}}}\frac{\delta {\mathbf{U}}_x}{{\mathrm{\partial }}_{\mathrm{n}}{\mathbf{U}}_x}\hspace{0.17em}\mathrm{d}\mathit{\Gamma }.$$ 4.36

The sensitivity of A_back to volume control obtained using (4.36) is shown in figure 8. Regions of large sensitivity extend from upstream of the bump summit all the way to the reattachment point, with opposite signs below and above the separatrix.

Figure 8.

Sensitivity of backflow and recirculation areas A_back, A_rec, with respect to streamwise volume control C_x. (a) Re=100 and (b) Re=500. The blue solid line is the separatrix, and the red dashed line is the curve where U_x=0.

(d). Recirculation area

We first rewrite the recirculation area (2.4) as

$$A_{\mathrm{rec}}={\iint }_{\mathit{\Omega }}{\mathbb{1}}_{{\mathit{\Omega }}_{\mathrm{rec}}}(x)\hspace{0.17em}\mathrm{d}\mathit{\Omega }={\int }_{x_{\mathrm{s}}}^{x_{\mathrm{r}}}{\int }_{y_{\mathrm{w}}(x)}^{y_{\mathrm{sep}}(x)}\mathrm{d}y\hspace{0.17em}\mathrm{d}x,$$ 4.37

where we recall that y_w(x) describes the wall height and y_sep(x) the separatrix height. Then we note that it is possible to give an Eulerian characterization of the separatrix, namely that the flow rate through any vertical cross section of the recirculation region is zero

$${\int }_{x_{\mathrm{s}}}^{x_{\mathrm{r}}}{\int }_{y_{\mathrm{w}}(x)}^{y_{\mathrm{sep}}(x)}{U}_x(x)\hspace{0.17em}\mathrm{d}y\hspace{0.17em}\mathrm{d}x=0.$$ 4.38

The sensitivity of the recirculation area with respect to flow modification is

$$\begin{array}{rl}\delta A_{\mathrm{rec}}& =\underset{ϵ\to 0}{lim}\frac{A_{\mathrm{rec}}(\mathbf{U}+ϵ\mathit{\delta }\mathbf{U})-A_{\mathrm{rec}}(\mathbf{U})}{ϵ}\end{array}$$ 4.39

$$\begin{array}{r}=\underset{ϵ\to 0}{lim}\frac{1}{ϵ}{\int }_{x_{\mathrm{s}}+ϵ\delta x_{\mathrm{s}}}^{x_{\mathrm{r}}+ϵ\delta x_{\mathrm{r}}}{\int }_{y_{\mathrm{w}}(x)}^{y_{\mathrm{sep}}(x)+ϵ\delta y_{\mathrm{sep}}(x)}\mathrm{d}y\hspace{0.17em}\mathrm{d}x.\end{array}$$ 4.40

Next, we use (4.38) to obtain the first-order variation of the separatrix height

$$\delta y_{\mathrm{sep}}(x)=-\frac{1}{U_x(x,y_{\mathrm{sep}}(x))}{\int }_{y_{\mathrm{w}}(x)}^{y_{\mathrm{sep}}(x)}\delta U_x(x)\hspace{0.17em}\mathrm{d}y.$$ 4.41

Substituting into (4.40), splitting integration intervals into [y_w,y_sep]∪[y_sep,y_sep+ϵδy_sep] and [x_s+ϵδx_s,x_s]∪[x_s,x_r]∪[x_r,x_r+ϵδx_r], and keeping first-order terms finally leads to

$$\delta A_{\mathrm{rec}}=({\mathbf{\nabla }}_{\mathbf{U}}{A}_{\mathrm{rec}}\hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{U})={\int }_{x_{\mathrm{s}}}^{x_{\mathrm{r}}}\frac{-1}{{\mathbf{U}}_x(x,y_{\mathrm{sep}}(x))}({\int }_{y_{\mathrm{w}}(x)}^{y_{\mathrm{sep}}(x)}\delta {\mathbf{U}}_x(x)\hspace{0.17em}\mathrm{d}y)\mathrm{d}x.$$ 4.42

The sensitivity of A_rec to volume control obtained using (4.42) is shown in figure 8. As could have been expected, it is very similar to the sensitivity of the backflow area.

5. Results

(a). Sensitivity maps

Figures 6–8 have already presented the sensitivity of all quantities of interest (2.1)–(2.4) to volume control. This sensitivity information can be used to compute the effect of a small control cylinder of diameter d inserted in the flow at (x_c,y_c). This effect is modelled as a steady volume force opposed to the hypothetical drag force the control cylinder would feel if it were invested by the uniform flow U(x_c,y_c)

$$\mathit{\delta }\mathbf{C}(x,y)=-\frac{1}{2}d{C}_{\mathrm{d}}(x,y)\parallel \mathbf{U}(x,y)\parallel \mathbf{U}(x,y)\delta (x-x_{\mathrm{c}},y-y_{\mathrm{c}}),$$ 5.1

where C_d is the cylinder drag coefficient. Its value depends on the local Reynolds number Re_d(x,y)=∥U_b(x,y)∥d/ν, which we compute from a fit of experimental and numerical data [15,19]. From (3.3) and (5.1), quantities of interest vary according to

$$\delta \varphi =({\mathbf{\nabla }}_{\mathbf{C}}\varphi \hspace{0.17em}|\hspace{0.17em}\mathit{\delta }\mathbf{C})=-\frac{1}{2}d{C}_{\mathrm{d}}(x_{\mathrm{c}})\parallel \mathbf{U}(x_{\mathrm{c}})\parallel {\mathbf{\nabla }}_{\mathbf{C}}\varphi (x_{\mathrm{c}})\cdot \mathbf{U}(x_{\mathrm{c}}).$$ 5.2

Note that the predicted value δϕ varies linearly with the force, by construction, but nonlinearly with the diameter of the control cylinder. Figure 9 shows the effect of a cylinder of diameter d=0.05 at Re=500. The separation point and separation angle are mostly affected if the cylinder is inserted close to x_s, and hardly vary otherwise. The reattachment angle is sensitive close to x_r and is weakly increased if the control cylinder is located in the shear layer. Overall, these three quantities appear robust because they cannot be modified easily (scales next to colour bars confirm that their variations are of small amplitude). The reattachment point is much more sensitive and is predicted to move downstream if the control cylinder is inserted in the shear layer (particularly at the bump summit) or upstream, and should instead move slightly upstream for a cylinder further away from the wall. Backflow and recirculation areas are affected in a fairly similar way, increasing when the control cylinder is located near the bump summit or upstream, and decreasing when the cylinder is further above the bump or the early recirculation region.

Figure 9.

Effect of a control cylinder of diameter d=0.05 on separation and reattachment angles α_s, α_r, stagnation points x_s, x_r, backflow area A_back and recirculation area A_rec. Re=500. The blue solid line is the separatrix, and the red dashed line is the curve where U_x=0. The black circles show the position of the control cylinder (x_c,y_c)=(25.7,2.5) used for validation in §5b.

Figure 10 shows the sensitivity to wall control. Arrows point in the direction of positive sensitivity. All quantities are significantly more sensitive to normal actuation than to tangential actuation (axes are to scale, so that arrows show the actual orientation relative to the wall). Separation and reattachment angles are naturally most sensitive close to x_s and x_r, respectively. More interestingly, α_s is also sensitive upstream of the bump and α_r at the bump summit and in the whole recirculation region. The separation point x_s is sensitive only at the bump summit. Finally, x_r, A_back and A_rec are efficiently controlled by wall actuation at the bump summit and to a lesser extent in the whole recirculation region; unsurprisingly, these three quantities have very similar sensitivities. Note that the sensitivities of α_s and x_s are very large at x_s, and the sensitivity of α_r is very large at x_r; for clarity, the longest arrows at these locations are not shown.

Figure 10.

Sensitivity of separation and reattachment angles α_s, α_r, stagnation points x_s, x_r, backflow area A_back and recirculation area A_rec to wall control U_c. Arrows point in the direction of positive sensitivity. The dashed line is the separatrix. Re=500.

Figures 9 and 10 allow us to identify regions where quantities of interest are affected the most by control and to what extent.

(b). Validation and control

In this section, we illustrate how control configurations can be designed based on sensitivity information. We also validate the method by comparing sensitivity predictions against nonlinear results obtained from actually controlled flows at Re=500.

Figure 11 shows how separatrix angles, stagnation locations and backflow and recirculation areas vary when applying small-amplitude vertical wall suction (U_c=0,V _c<0) over 5≤x≤23, with total flow rate W. All quantities decrease, although not by the same amount: the reattachment point moves significantly upstream, inducing a large reduction in backflow and recirculation areas. Separatrix angles decrease only slightly. The separation point is virtually fixed, reminiscent of the fact that it is fairly independent of Re in the uncontrolled case (figure 4). The agreement between sensitivity predictions (straight solid lines) and actual results (symbols) is excellent at small flow rate. However, nonlinear effects are non-negligible when $|W|\gtrsim 0.1\text{--}0.2$ and, in all cases, make actual variations smaller than predicted by sensitivity analysis.

Figure 11.

Effect of wall suction applied over 5≤x≤23 with total flow rate W, at Re=500. (a) Variation of characteristic separation quantities (recirculation and backflow areas, stagnation locations, separation and reattachment angles). Thick solid lines show theoretical predictions from sensitivity analysis, while symbols show results from nonlinear calculations. (b) Separatrix for the uncontrolled flow (W=0, thick line), and W=−0.2, −0.4 and −0.6 (thin lines). (Online version in colour.)

Figure 12 shows variations of quantities of interest when a small control cylinder of diameter d=0.05 is inserted in the flow at x_c=(x_c,y_c)=(25.7,2.5) (this location is shown in figure 9 with a black dot). All quantities increase with the cylinder diameter, especially the reattachment point and recirculation and backflow areas, whereas the separatrix angles and separation point are less affected. Again, nonlinear effects are observed when $d\gtrsim 0.05$. This is consistent with the nonlinear variations measured by Parezanović & Cadot [13] with d=1 and 3 mm. Recall that, in addition to the nonlinear effects neglected by our linear sensitivity analysis, variations (5.2) are linear in δC but not in d (as the drag coefficient depends on d), therefore sensitivity predictions in figure 12 are not straight lines, unlike those in figure 11. We also report results obtained with the control cylinder included in the computational mesh (grey filled symbols) in order to assess the assumption of uniform flow underlying (5.1). Differences can be noted when $d\gtrsim 0.05$, but sensitivity analysis does provide useful qualitative information regarding the influence of small passive control devices.

Figure 12.

Effect of a small control cylinder of diameter d inserted in the flow at (x_c,y_c)=(25.7,2.5), at Re=500. (a) Variation of characteristic separation quantities (recirculation and backflow areas, stagnation locations, separation and reattachment angles). Thick solid lines show theoretical predictions from sensitivity analysis, while open symbols show results from nonlinear calculations, both with the force model (5.1). Grey filled symbols show results from nonlinear calculations with the control cylinder included in the mesh. (b) Separatrix for the uncontrolled flow (thick line) and for d=0.02, 0.04 and 0.06 (thin lines). (Online version in colour.)

6. Conclusion

Considering the boundary layer flow above a wall-mounted bump as a prototype for separated flows, a variational technique was used to derive analytical expressions for the sensitivity of several geometric indicators of flow separation to steady actuation: the locations of the two stagnation points (x_s and x_r), the angles of the separatrix (α_s and α_r) at these points, the backflow area A_back and the recirculation area A_rec. For each geometric quantity, analytical expressions for the linear sensitivity to base flow modification ∇_U^* obtained. This gradient information was further translated in sensitivity maps to localized volume forcing and wall blowing/suction through the introduction of the adjoint base flow, governed by linear adjoint equations forced by the previously determined gradient ∇_U^*. A suitable modelling of the addition of a small control cylinder as a localized force depending on the local velocity allowed us to obtain sensitivity maps relevant to experimental studies.

Validations against full nonlinear Navier−Stokes calculations showed excellent agreement for small-amplitude control for all considered indicators. Non-linear effects appeared at larger amplitudes, consistent with experimental observations. With very resemblant sensitivity maps, the reattachment point, the backflow and recirculation areas were seen to be easily manipulated. By contrast, the upstream separation point and the separation and reattachment angles were found to remain extremely robust with respect to external steady actuation.

The present analysis, however, is limited to steady actuation and calls for a generalization to the sensitivity of mean recirculation properties to harmonic forcing, which is known to be a more realistic, reliable and efficient experimental control scheme. Additionally, the recent development of fast imaging techniques has now made these geometric descriptors accessible in real time [6], highlighting the need for the generalization of current open-loop control optimization tools to the dynamic closed-loop control of separation. The recent development of a solid theory for unsteady separation [3,29] provides a firm grounding for this challenging objective.

Funding statement

This work was supported by the Swiss National Science Foundation (grant no. 200021-130315) and the French National Research Agency (project no. ANR-09-SYSC-001).