1 Introduction

We consider two-dimensional flows whose sample thicknesses are much smaller than the lateral extensions so that their geometrical configurations can be taken as two-dimensional surfaces embedded in the three-dimensional Euclidean space. The fascination of two-dimensional flow comes from its potential application in, say, geophysical flows^[1], bio-membrane flows^[2], combustion, coating problems and diverse flow interfaces^[3]. Its interest also lies in the fact that it is itself an application of the methods and ideas which have been developed for the non-Euclidean space^[4].

On one hand, the two-dimensional flow, even in its trivial planar form, has many defining features that significantly differ from three-dimensional flows. For instance, flows in two dimensions lack the mechanism of vortex stretching, which implies that both energy and enstrophy are conserved^[5]. In addition, inverse energy cascade can be observed in two-dimensional turbulence. These can be illustrated by many experiments on soap flows concerning the associated physics^[6-9], two-dimensional turbulence^[10-11], and solid-fluid interaction^[12].

On the other hand, two-dimensional films can be easily bent so that the surface geometry may play a key role in the in-surface behavior. The curved case is more commonly seen in real applications. The interplay between geometry and internal structure is well studied for condensed matter systems^[13]. Technologically, such defects can be important in the design of soft materials^[14]. For fluid films, the study of interplay between geometry, especially the curvature, and the flow behavior is relatively less explored^[15], although it can be dated back to 1902 in Ref. [16] which analyzed in detail the vortex motion on a sphere. The vortices as surface defects and its interaction with geometry are studied for super fluidic films in Ref. [17]. Those with viscosity are studied in Refs. [15] and [18]. Less explored is the non-singularly distributed vorticity field on the curved surface. In a more general situation, the flow field may be drastically affected by the surface deformation^[19-21], and even vortices can be generated by oscillation of soap film^[22], which usually involves more complex geometric effects and the coupling with other media surrounding it.

Extensive work has been done on the modeling of two-dimensional flows. It includes conducting thin film approximations based on the three-dimensional Navier-Stokes equation^[23-26] and from the viewpoint of two-dimensional continuum mechanics^{[2-4, 27-29]}. In this study, we adopt the second kind of modeling and the deformation theory proposed in Ref. [29]. We focus on incompressible viscous flows on fixed surfaces whose governing equations are featured by the appearance of Gaussian curvature and surface metrics. Much attention will be focused on locally curved surfaces to explore quantitatively how the vorticity is generated by the Gaussian curvature, and how the existed vortices can be affected by the geometry, based on numerical results. Like Ref. [15], here we are interested in the effect of an imposed surface geometry on a relatively simple flow field, aiming at understanding the underlying mechanism.

Besides, vorticity dynamics is of crucial importance in the entire fluid mechanics. Vorticity dynamics of three-dimensional flows has been extensively studied and reviewed in Ref. [30]. Some of those important theoretical conclusions were extended to the case of two-dimensional surface flows in Ref. [31]. Therefore, in the present investigation, we hope to see the characteristics in concrete numerical examples from the prospective of vorticity dynamics.

The paper is organized as follows. In Section 2, some fundamentals of incompressible flows on the curved surface and the vorticity-based numerical method are introduced. Section 3 presents the analysis of a free flow over the locally curved surface to see the mechanism of vorticity generation by the Gaussian curvature. In Section 4, the interplay between cylindrical wake and surface geometry is investigated. Summary and future directions are provided in Section 5.

2 Governing equations and computational overview 2.1 Elementary differential geometry of surface

We assume a general two-dimensional flow on a fixed curved surface. The surface can be parameterized as , with x = {xⁱ}, i =1 and 2 being the parametric/curvilinear coordinates. The covariant basis vectors g_i, i =1 and 2, the unit normal vector n_Σ, the metric tensor G=[g_ij], the curvature tensor K = [b_ij], the Gaussian curvature K_G, and the mean curvature H are defined as

Einstein summation convention is adopted throughout the paper with subscripts and superscripts representing co-and contra-variant quantities respectively taking integers from 1 to 2. We use elementary of modern differential geometry, whose classic materials can be found in, e. g., Refs. [31] and [32].

2.2 Governing equations of dynamical process

Denote the surface density of the film as ρ, which satisfies ρ = hρ_v (ρ_v is the bulk density and is assumed to be constant). ρ can be variable due to the variation of the thickness h. Thus, viewed as a two-dimensional surface continuum, the flow can be classified as incompressible or compressible in terms of the surface density. By using the surface gradient operator and the Levi-Civita gradient operator ∇ discussed in detail in Ref. [29], and following the theoretical model proposed in Ref. [31], the governing equations for the incompressible (neglect the thickness variation) surface flow read

(1)

(2)

where ∇_l and ∇^l := g^lk∇_k denote the Levi-Civita connection^[31]. μ denotes the surface viscosity and γ the surface tension. We have decomposed the external force f and the acceleration a into the tangent and normal parts, . For a fixed surface, the in-surface flow would not be affected by the normal force, and hence it is decided totally by the tangent part of the momentum equation, whose component form is

(3)

The tangent part of the above governing equation is in accord with the one used in Ref. [15].

2.3 Vorticity-stream function formulation for incompressible flows on surfaces

We now give the vorticity-based formulation for solving incompressible viscous flows on fixed surfaces, which is a good approximation of a big class of real flows. Based on the continuity equation (1), there is

where g := det(g_ij). A stream function ψ is introduced through . Define the surface vorticity as . Subsequently, a surface vorticity-stream function formulation can be derived,

(4)

where Re := ρV_∞/µ, V_∞ denotes the far-field incoming velocity, Fr := V_∞² /(g₀L), and g₀ denotes the characteristic unit external force. Based on the intrinsic generalized Stokes formula of the second kind^[29], one has

This indicates that the integral dl is independent of the integral paths. Therefore, the stream function along a boundary curve can be determined through

(5)

where r₀ represents the reference position arbitrarily chosen on the surface boundary.

Though the pressure p (= γ) is not involved in the vorticity-stream function formulation, it can be solved either with the known velocity field

or by integration of (3).

2.4 Computational overview and numerical method

The dynamics of vorticity will be illustrated by two numerical examples below, including the free flow and the cylindrical wake on a locally curved surface. In the analysis, we will also present the planar case of cylindrical wake for comparison. The governing equations are numerically solved with the finite difference method. We use the general curvilinear coordinates so that the mesh grid is body fitted. The physical domain is mapped onto a rectangular parametric domain by a smooth mapping. Therefore, the differential equations are discretized on the parametric domain under local basis. Spatial derivatives are discretized using central difference. The flow is advanced in the time using the third-order Runge-Kutta method. The details of vorticity based on the numerical method in curvilinear coordinates can be referred to Ref. [33]. The resolution is chosen such that the calculations repeat with no significant change of the results with finer meshes. We have checked that our code could well simulate the classical incompressible planar cylindrical wake.

3 Free flow over locally curved surface: generation of vorticity

We first investigate a simple case to illustrate that the Gaussian curvature can generate vorticity on the surface. A free flow over a locally curved surface at Re=100 is studied. A hill bump described by z(x, y) = h exp(−(x² + y²)/(2r₀²)) is located downstream, where x is the streamwise coordinate, and y is the spanwise one. The bump shape depends on two parameters: the height of the hill h and the half decay radius r₀. The parallel incoming velocity is set as V_∞ = 1.0. The Re number is based on V_∞ and h. A group of flow simulations are carried out under various parameters. Here, we mainly exhibit the case when h=2 and .

3.1 Basic properties of flow field

Figure 1 shows there are two bands of vorticity generated behind the bump, which is antisymmetric about y=0. The sign of vorticity at y < 0 (y > 0) is negative (positive), which is opposite to the case of a cylindrical wake. Though the magnitude of vorticity is comparatively small, the patterns do reveal the role of surface curvature which will be discussed in detail. Figure 2(a) plots the vorticity profiles along different spanwise curves. The vorticity anti-symmetrically occurs near the curved region and is strengthened on the bump. There is a significant decrease in the velocity in the curved region (see Fig. 1(b)), as also shown in Fig. 2(b), which is due to the fact that the curve length is enlarged at the bump.

One may doubt why the sign of vorticity distribution suggests a velocity jet near the center line while actually there is a velocity deficit? Meanwhile, the vorticity generated in the field is quite small in the magnitude. We will show that the visible velocity deficit is actually caused by the surface geometry, more precisely, the surface metrics. To illustrate that, we calculate the velocity field of a potential flow under the same boundary conditions, whose stream function satisfies ∆ψ=0. Considering that the velocity circulation is everywhere zero in a potential field, and the velocity flux at any spanwise cross-section (spanwise curve) keeps constant, a decrease in the velocity on the bump is expected. We denote the velocity of the original viscous free flow as V and the one of the potential flow as V_p. Figure 3 gives the profiles of velocity difference between these two cases at different cross-sections. Compared with the velocity norm, which is of order O(1), the difference is less than 1%. However, it is this tiny difference that forms the vorticity bands behind the bump.

By subtracting the potential field from the original one, we actually obtain the potential-free flow. Figure 3(b) clearly shows that there is an increase in the velocity near the center layer, which leads to the velocity shearing and forms the vorticity on both sides. The sign of the vorticity now makes sense according to the velocity shearing. This increase in the velocity can be explained by the driven force term μK_GV in the momentum equation. Therefore, it can be expected that a negative K_G could accelerate the velocity compared with the planar case, causing an additional decrease in the velocity near the center layer compared with the potential flow. To validate this prediction, a more complicated surface with the prominent negative Gaussian curvature is designed as z(x, y) = h(exp(−x²)−exp(−y²)) exp(−c(x² +y²)), x = (x−y)/2, y = (x+y)/2 with h = 2, c = 0.25 (see Fig. 5(a)). Figure 4 presents the vorticity field and the stream-lines with velocity norm distribution. Comparing Fig. 4(b) with Fig. 5(a), we see a clear link between curvature and velocity, which is caused by surface metrics and will also hold for the inviscid case. As indicated in Fig. 4(a), although the flow field pattern is more complicated, there tends to be an upper-red-lower-blue vorticity distribution after each positive K_G region and a converse vorticity band after the dominant negative K_G region. A possible explanation in terms of driven and drag force caused by μK_GV is illustrated in Fig. 5(b).

Since the bump is rotationally symmetric, it might be more intrinsic to use the φz-coordinate (circumference-height) [x, y, z](φ, z)=[R(z) cosφ, R(z) sinφ, z] in the curved region, where R(z) is decided by the surface parametrization. The velocity components V_z and V_φ can be interpreted as the climbing velocity and circulating velocity, respectively. The vorticity can then be decomposed into three parts as , which can be termed as the shearing term of climbing velocity, shearing term of circulating velocity, and geometry contribution of curvilinear coordinates, respectively. Actually, from this decomposition based on the potential-free field, the computational results show that the vorticity is dominantly contributed by the V_{z, φ}R⁻¹(z) part.

3.2 Mechanism of vorticity generation and evolution

The above analysis of vorticity is based on the velocity shearing of the potential-free field. In this subsection, we investigate the mechanism of vorticity generation based on the vorticity evolution equation, which on a general curved surface reads

(6)

where is a key vector, behind which is the physical source, i.e., the shearing kinetics^[30]. -θω³ is a kinematic effect and vanishes in the incompressible case. For incompressible flows, the shearing kinetics term is determined by

(7)

Figure 6 gives the distributions of these terms. It shows that, on the upwind side of the bump, the geometric curvature term makes the main contribution to the vorticity source. In the downstream of the bump, the main physical source is the viscous dissipation of vorticity, opposite in the sign to the distribution of curvature term. Since the diffusion term works only in the presence of non-zero vorticity, to understand the generation of vorticity, we focus on the curvature term. In fact, it can be further expressed by two components,

It is revealed from Fig. 7 that, in this case, the curvature term is mainly determined by non-uniformity of Gaussian curvature (∇K_G × V) · n_Σ. If we view it in the φz-coordinate frame mentioned above, we may clearly see why the vorticity distribution shows a positive value in the upper field (y > 0). Now, (∇K_G × V) · n_Σ reads , the first term equals zero since K_{G, φ}= 0. In the second term, K_{G, z} is even in y, while V_φ is odd and negative when y > 0. Subsequently, an overall positive curvature term leads to a positive vorticity in the upper field (y > 0), and vice versa.

We now compare the contributions of , and ω³ along a streamwise line along which the maximum vorticity of the whole field occurs, as shown in Fig. 8. We can see that, before the crest, the curvature term makes the most contribution to the total vorticity source. Meanwhile, it can also be observed in Fig. 8 that, when , the vorticity increases, and vice versa. Then, we can conclude that the vorticity is mainly generated by the curvature term, more precisely, non-uniformity of Gaussian curvature. For developable surfaces, the Gaussian curvature vanishes, and the new vorticity generation does not play a role. In Fig. 9, a folded region (K_G = 0) is located downstream and has few effects on the vorticity evolution and velocity norm distribution. Let us sum up the process. Non-uniformity of Gaussian curvature causes the non-zero curvature term, which acts as the source of vorticity generation. After the crest, the main behavior of vorticity is determined by the diffusion process and transport process by the convective term.

3.2.1 Effect of surface parameter

Although the local generation of vorticity is due to non-uniformity of Gaussian curvature, the global evolution of vorticity field is a synthetic process by curvature, diffusion and convective processes. For example, Fig. 10(a) gives the distribution of vorticity sign over the flow field. The vorticity is generated by the curvature term. However, it does not mean that the vorticity would always keep positive or negative along the downstream direction. Therefore, it needs to be further studied based on the surface geometry and other factors.

We now examine the dependence of global behavior of flow field on the surface parameters. For the Gaussian bump, the Gaussian curvature as a function of the radius from the center point can be represented as . Note that α:=h/r₀ controls the overall magnitude of G(r). At r=r₀, G(r) changes its sign. In the present case, , and h=2. The distribution of K_G as a function of r is given in Fig. 10(b). is also plotted there, which has the same sign as in the φz-coordinate used above. It is worthwhile to mention that the profile of is quite close to that of vorticity shown in Fig. 2(a), which again illustrates the correlation between the surface geometry and vorticity generation.

Since the vorticity is band-like distributed in the far downstream field (see Fig. 10(a)), we now look at the width of the band and the critical spanwise position at which the sign of vorticity changes. It is found that the width is unchanged from about x=2.0 all the way to the far field. We call it the width of vorticity band. Figure 11(a) gives the width in various cases as a function of r₀. It is revealed from the fitting curve that, as r₀ increases, the width of vorticity band in the corresponding case increases linearly. This indicates that r₀ may decide the wake width. The maximum value of vorticity decreases as r₀ increases. The results tally with our intuition that under the same height, a gentle slope may cause a comparatively mild flow field. Figure 12(a) shows that the maximum value of vorticity is linear about 1/(2r₀²). Keeping the exponent part of the shape unchanged, we merely change the height from h=0.5 to h=2 and observe the flow field. From Fig. 11(b), we can see the height variation makes few contributions to the width of vorticity band, the downstream vorticity band only narrows slightly as the bulge becomes higher. However, the value of the maximum vorticity takes a prominent linear increasement with h (see the fitting curve drawn in Fig. 12(b)). Figure 13 shows the comparison of flows under various bump heights by vorticity profiles at different streamwise positions. We can see that, at x=0, the distributions are linearly dependent on h, while this is no longer the case in the downstream area. This is in accord with the above observation that the generation of vorticity is closely related to the geometry, and its evolution after that is a comprehensive process.

3.2.2 Effect of Reynolds number

Now we investigate the effect of Re on the generation of vorticity. Figures 11(c) and 12(c) show the dependences of maximum vorticity and wake width on the Re. The maximum vorticity approximately shows linear dependence on 1/Re. The generation of vorticity is mainly due to the curvature term, which is multiplied by 1/Re. The Laplace term also includes 1/Re, but it mainly takes part in the diffusion process, influencing indirectly the maximum vorticity. However, Re has no effect on the wake width. This further confirms that it is the geometry (r₀) that determines the width of vorticity band.

Figure 14 shows the comparison of flows under various Reynolds numbers by vorticity profiles at different streamwise positions. In general, the profiles under various Re are similar to each other. The generated vorticity decreases in the magnitude as Re increases. The distribution shows less change in the magnitude when Re gets higher. In the downstream, the differences are less evident than those near the bump. The widths of vorticity bands, reflected by the length of positive/negative part of the profiles, are nearly the same in each case.

4 Cylindrical wake over locally curved surface: disruption of vortex street

The above free flow case illustrates how the vorticity is generated by the distribution of Gaussian curvature. The vorticity is small but wholly caused by the Gaussian curvature rather than by the boundary rubbing. The surface metrics does have effects, but mostly on the irrotational field of velocity, resulting in a decrease in the velocity in the curved region. Below we study the wake of a circular cylinder in which vortices are already formed before passing the curved region. We would like to see how the wake would be changed by the surface geometry and what is the cause.

4.1 Basic properties of flow field

Figures 15 and 16 give the vorticity distribution of the wake of a circular cylinder at one instant. The flow is confined on a locally curved surface. A hill bump described by z=h exp(−((x − x₀)² + (y − y₀)²)/(2r₀²)) is located downstream (x₀=6.0, y₀=0.5, h=1.6, and 1/(2r₀²) =0.4). Here, z is the height as a function of x and y, x is the streamwise direction, and y is the spanwise one. The cylinder is placed at (0, 0) with its radius r=1. The flow is incompressible, and Re=500 is based on the diameter of the cylinder. Since the position of the hill is not located on the center line, the wake is different from the case of a planar flow, with a third row of vortices occurring beside the common vortex sheet. The local flow structures are also disturbed due to the hill. In contrast with the planar case, the upper two rows of vortices in the far wake are nearly side by side rather than interlaced. Figure 15 also gives the streamlines of the flow with the background being the contours of velocity norm for the curved and planar surface case. It can be seen that the third row of vortices is transported faster than the other rows, which explains the larger spacings among the vortices in this row.

To analyze the flow field, the vorticity evolution equation is first examined. The contributions of (∇ × a) · n_Σ and the convective term at one instant are given in Fig. 17. It can also be observed that the overwhelming contribution of (∇ × a) · n_Σ is the viscous dissipation term Re^-1∆ω³, while the curvature terms are negligible in the magnitude compared with it. This is different from what we have seen in the free flow case, in which the main contribution is the Gaussian curvature term. It is reasonable since in the previous case, the flow field upstream is almost vorticity-free, and the Gaussian curvature gives rise to the generation of vorticity, while at present the dominant flow structure is the vortex street formed by the strong shearing process around the cylinder. We have discussed the role of Gaussian curvature term in the process of vorticity generation. Actually, it is also involved in the viscous diffusion process. The term K_Gω³ may strengthen the norm of vorticity when K_G > 0 and weaken it when K_G < 0. This will work when the existed vorticity is transported over the region with the non-zero Gaussian curvature, especially when K_G is large. In the present case, it is found that when vortices pass the bump, the curvature related terms are of order 10^-2, whereas that of the main diffusion term Re^-1∆ω³ is about 10^-1. Now that the curvature related terms are relatively small, and the vortex street is indeed changed by the bump, we need to further investigate the flow field to reveal the possible cause. Since the flow field is unsteady, the transport process should be paid attention to.

Figure 18 shows stream-lines with local vorticity distributions and the corresponding velocity norm distributions at successive instants, covering half of a vortex shedding cycle. A distribution of the principal stretching rate with its eigen direction is also provided. Apparently, the region where the vorticity is large is always accompanied by the small velocity. The patterns show a great similarity between the vorticity and the principal stretching rate, and the principal direction lines always pass through the vorticity region. Look into the vorticity evolution process in Fig. 18, and we may find that the vortex looks like being pressed on the upslope of the bump and then being elongated, allowing the vortex to rotate and break into two rows downstream. This implies that the transport of the vortex is slowed down on the upwind side of the bump. We recall that in the free flow case in Section 3, there is a significant velocity depression in the bump region which is due to the surface metrics.

Considering this, a more quantitative examination is shown in Fig. 19. Comparing the present case in which a bump is located at y=0.5 with the planar case, we plot in Fig. 19 the streamwise velocity profiles along different spanwise lines at one instant when they are experiencing the same vortex shedding phase (their shedding periods are almost the same). It can be observed that, in the upstream region of the bump, the velocity profiles are similar to each other, with only a tiny difference. However, for the bump case at x=5 (on the upwind side of the bump), the velocity profile undergoes a prominent decrease compared with the planar case. Therefore, a vortex on the upwind side of the bump is tending to be pressed along the streamwise direction. At downstream, say x=10, the streamwise velocity profiles recover to a similar level to the planar case, as shown in Fig. 19(c). In summary, although the velocity distribution is more complex than that of the free flow case, there still exists a significant velocity depression in the bump region compared with the planar cylindrical wake case, which leads to the pressing of vortex passing by.

4.2 Geometry effects on vortex evolution and transportation

In fact, there are two important stages in the formation of the third row of vortices. One is the stretching of the vortex passing over the bump. As illustrated above, the bump causes a velocity depression, which will slow down the transport of vorticity passing there. Figure 20 gives the velocity and vorticity profiles along y=0.5 at different time, compared with the planar case.

At each time considered in Fig. 20, the planar case and bump case are undergoing the same vortex shedding stage. However, due to the existence of the bump, the velocity profiles become quite different in downstream regions. Since (6, 0.5) is the peak of the bump, the velocity around there in the bump case is significantly smaller than the planar case. Consequently, the vorticity transport is slowed down there, compared with the planar case. In Fig. 20, the vorticity profile of surface flow within x∈[4, 4.5] is similar to the planar case t=487.5, and in upstream regions they are almost the same. While t=489.5, the vorticity peak in the planar case is distinctly ahead of that of the bump flow. Meanwhile, the non-zero vorticity region in the planar case has been much wider than the later one. This suggests that the vorticity transport is pressed in front of the bump. Their maximum values of vorticity are similar, indicating that a narrower streamwise distribution will be accompanied by a spanwise stretching, considering that the total amounts of vorticity in both cases are decided by the wall-boundary shearing, and they are similar. In order to show this process more clearly, Fig. 21 gives the local vorticity distributions near the bump at different time compared with the planar case. At t=489.5, the negative vortex is significantly bent near the center of the bump and also looks thinner compared with the planar case.

The other stage is the disconnection of the elongated vortex. The separated vortices flow into the first and third rows of vortices individually, as shown in Fig. 22. In Fig. 22(a), the negative vortex is continuously distributed, while in Fig. 22(c), it has been cut off during the rotation of the upper part. After that, the lower part will further sweep downstream and be fully separated by the positive vortices (the second row). We will show below that the first and third rows of vortices will then evolve in individual ways and transport at different velocities.

In conclusion, the geometry affects the vorticity dynamics from two aspects, one is the Gaussian curvature term explicitly taking part in the governing equations, and the other is the surface metrics. In the free flow case, the Gaussian curvature term plays the dominant role in the generation of vorticity while the latter one mainly has effects on the irrotational part, since the background field is basically irrotational, and the generated vorticity is the main vortical structure in the flow. On the contrary, in the present cylindrical wake case, the Gaussian curvature term itself has no significant explicit effect on the flow, whereas the velocity depression caused by the surface metrics significantly distorts the transport of vortices after being formed, leading to the third row of vortices. The surface metrics, though it does not seem as explicitly displayed as the Gaussian curvature term, indeed plays an important role in determining the vorticity dynamics on the surface, no matter in inviscid flows or viscous flows. In Refs. [16] and [17], which consider the inviscid flow (the point vortices system, K_G does not explicitly take part in the governing equations), the flow is greatly changed due to the surface geometry.

4.3 Spatio-temporal analysis of flow field

After looking into the near wake structures, we now investigate the time history of the flow field to further see how the third row of vortices is reflected. Figure 23 shows the auto-power spectra at various locations with respect to V_x and ω³. We choose spanwise positions corresponding to each row of vortices. Unlike the planar case, since the positive (red) and negative (blue) vortices are no longer symmetrically located, f₀ becomes the dominant frequency at all these positions. For ω³ and V_x, harmonics are more obvious at the third row than other positions. There are no evident frequency components other than harmonics although the third row vortices exhibit increased spacial intervals. This indicates that it is the cylinder that generates the temporal periodic structures, and the surface geometry only contributes to the spacial redistribution of the generated vortical structures.

The third row vortices exhibit the same dominant frequency as those of the other two rows, but have larger vortex intervals. In order to see their evolution more clearly, the phases associated with the main frequencies are investigated. In the flow stability analysis, the spatial evolution of velocity fluctuation is usually supposed to take the form of traveling waves as , where A_k(f_m, r) is the amplitude corresponding to the frequency f_m as a function of position vector r, α(f_m) is the wave number vector and could be a complex number if there is exponential decaying in the space, ϕ₀(f_m) is the initial phase, and k=1 and 2 correspond to the components of velocity fluctuation in the Cartesian space. The above Fourier series can also be expanded as , where C_k(f_m, r) = A_k(f_m, r) · exp(i(α(f_m) · r + φ₀(f_m))). The generalized cross-spectra S(f_m; r, r₀) can therefore be expressed as S(f_m; r, r₀) := C_k(f_m, r) · C_k^∗(f_m, r₀). If α_R denoted as the real part of α is independent of r, then the phase of cross-spectra has the form arg(S(f_m; r, r₀)) = α_R(f_m) · (r − r₀). Thus, we can calculate the phase difference arg(S_xx0(f_m, x)) at f_m between two streamwise positions r = (x, 0, 0) and r₀ = (x₀, 0, 0), with the latter one being a fixed reference probe. If the phase difference is linearly dependent on x, it suggests that the spatial evolution has a constant streamwise wave number α_x(f_m) which can be obtained by the slope, and a constant phase-velocity c_ϕ = 2πf_m/α_x corresponding to f_m. In the shear layer of a circular jet flow, the vortex pairing and merging are associated with coincidence of the wave velocities corresponding to the dominant frequency and its first order subharmonic component, which is called the wave velocity matching or resonance^[34-35]. It is found in the experimental study of circular cylindrical wake at the low Reynolds number that the wave number with respect to some range of frequency keeps the same, which is called the wave number matching or resonance^[36].

Figures 24 and 25 give the spatial evolution of phase of cross-spectra along y=-0.5 and y=-4.6 with respect to ω³, V_x, and V_y. The streamwise lines y=-0.5 and y=-4.6 correspond to the second (positive) and third (negative) rows of vortices, respectively. It is found that the phase difference depends linearly on the streamwise position for f₀ and its harmonic components. This indicates that there is a constant x-direction wave number for each frequency, being different from each other since the slopes are different. The constant wave velocities c_ϕ are summarized in Table 1, corresponding to different frequencies and variables. We find that no matter for the second or the third row (y=-4.6), the wave velocities at these frequencies are close to each other, indicating few dispersion effects. This holds for both vorticity and velocity. However, the wave velocity at y=-4.6 is much larger than that at y=-0.5 and even the far field velocity. This is in accord with the velocity distribution (see Fig. 15). This also clearly illustrates in terms of frequency-domain analysis that the larger intervals in the third row are caused by faster transport. With the obtained wave velocities c_ϕ (approximately 0.76 for the second row and 1.1 for the third row) and the prime frequency f₀, we can calculate the wave length λ_w=c_ϕ / f₀, namely, the vortex intervals in each row. For the second and the third rows, λ_w is about 7 and 10, respectively, both of which agree well with the distances between two neighboring vortices shown in Fig. 15.

We have also studied the effect of the bump position. In the above case, the bump center is placed at y₀=0.5, and the third-row vortices still exist when y₀=1.0 (see Fig. 26(a)). If y₀ is far from the wake region, the influence would be negligible (see Fig. 26(a)), so does the case y₀=0. Besides, we have also considered various flow parameters. Though not the main aim in this paper, we list some of the results.

(Ⅰ) Effect of Re

At the relatively high Re, this effect is more obvious, and when Re=100 the third row of vortices is relatively weak.

(Ⅱ) Effect of Fr

The external force plays an important role in keeping flow configuration. For a fixed surface, only the tangent external force has effects on the in-surface flow, and the normal force merely takes part in maintaining the surface shape. We consider the tangential friction as widely adopted in study of two-dimensional turbulence^[10], which leads to αω³ in the vorticity equations. We find that an increasing surface friction may prevent the shedding of vortex pairs. When the surface friction coefficient α reaches the order of 10^-1 at Re=500, the flow will be steady, and no vortex sheet will be formed in the wake (see Fig. 27(a)). The attached vortices are shorter for a higher α (see Fig. 27(b)).

(Ⅲ) Effect of geometry parameters

The geometry parameters of the bump may affect the flow from two aspects. On one hand, the aforementioned pressing mechanism would be more obvious if the surface is higher or steeper. On the other hand, the effect of vorticity generation mechanism caused by the Gaussian curvature may be enhanced when K_G gets larger. Thus, the stronger interaction of these two aspects might make the evolution even more complicate.

5 Conclusions

Incompressible viscous flows on curved surfaces are considered with respect to the interplay of the surface geometry, curvature, and vorticity dynamics. Numerical results show that non-uniformity of Gaussian curvature could generate the vorticity. In the cylindrical wake, the surface geometry could significantly affect the vortex street by forming a velocity depression. The results may provide possibilities for manipulating surface flows through local change in the surface geometry. In the present work, we consider the Gaussian bump which dominantly possesses the positive curvature over the curved region. We aim to analyze the mechanism of vorticity generation by curvature and vortical structure evolution affected by the surface geometry. For the negative curvature, the mechanism can be revealed similarly according to the above analysis. However, locally curved surfaces with the overwhelmingly distributed negative curvature are not common, always accompanied by appearance of the significant positive region. Hence, the situation will be complicated. Here, we focus on the imposed geometry. However, in many practical cases, the surface is under deformation and involves coupling with the medium around it, and the surface thickness is probably variable. Some basic theoretical results regarding vorticity dynamics on deformable surfaces and compressible (with varying surface thickness) flows are considered in our ongoing work.