Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds

2.1 Surfaces

The task is to solve a boundary value problem on an arbitrary surface Γ in three dimensions. Let the surface be fixed in space over time, possibly curved, sufficiently smooth, orientable, connected (so that there is only one surface), and feature a finite area. There is a unit normal vector on Γ. The surface may be compact, ie, without a boundary, ∂Γ=∅, see Figures 1A and 1B for examples. Otherwise, it may be bounded by ∂Γ, as shown in Figures 1C and 1D. Then, associated with ∂Γ, there is a tangential vector t_∂Γ pointing in direction of ∂Γ and a conormal vector n_∂Γ=n_Γ×t_∂Γ pointing “outwards” and being normal to ∂Γ and tangent to Γ. The surface may be given in parametrized form or implied, eg, based on the level-set method; both situations are considered herein. For the equivalence of these two cases and more mathematical details, see, eg, the work of Dziuk and Elliott.18

2.2 Surface operators

2.2.1 The tangential projector

On the manifold Γ, the tangential projector is defined by the normal vector as

Some important properties are (i) P·n_Γ=0, (ii) P=P^T, and (iii) P·P=P.

2.2.2 Surface gradient of scalar quantities

The tangential gradient operator ∇_Γ of a differentiable scalar function on the manifold is given by

(1)

where ∇ is the standard gradient operator and is a smooth extension of u in a neighborhood of the manifold Γ. Of course, may also be some given function (rather than an arbitrary extension) in global coordinates, ie, . For the case of parametrized surfaces defined by the map and a given scalar function , the tangential gradient may be determined without explicitly computing an extension using

(2)

with being the (3×2)-Jacobi matrix and G=J^T·J being the metric tensor (first fundamental form). Equation 2 shall be used later in the context of the FEM to determine tangential gradients of shape functions. It is noteworthy that ∇_Γu is in the tangent space of Γ and, thus, P·∇_Γu=∇_Γu and ∇_Γu·n_Γ=0. The components of the tangential gradient are denoted by

representing the first-order partial derivatives on Γ. Second-order partial derivatives may be denoted by

where is the tangential Hessian matrix. In the context of manifolds, this matrix is not symmetric,35 ie, for mixed second derivatives for i≠j.

2.2.3 Surface gradient of vector quantities

Next, operators for vector quantitites are considered. The “directional gradient” of u is the tensor of tangential derivatives and defined as

In contrast, the covariant derivatives are

One has to carefully distinguish these two different gradient operators. It is noted that appears frequently in the modeling of physical phenomena on manifolds, ie, in the governing equations. On the other hand, is often used in straightforward extensions of identities such as product rules and divergence theorems. For example, we have for a scalar function f(x) and vector functions u(x), v(x)

However, the relations are less straightforward for the covariant counterparts and , respectively. Later on, in the context of FEM implementations, it proves useful to transform covariant derivatives systematically to directional ones. This allows the computation of directional derivatives of FE (finite element) shape functions with respect to independent of the integration of the weak form of the governing equations.

2.2.4 Divergence operators and divergence theorem

The divergence of a vector function is given as

For a tensor function , there holds

It may be shown that div_ΓP=−ϰ·n_Γ with ϰ=tr(H) being the mean curvature and being the second fundamental form.

The following divergence theorem on manifolds is later needed for deriving the weak forms35, 36:

(3)

where . For tangential tensor functions with A=P·A·P, the term involving the curvature ϰ vanishes because then A·n_Γ=0. In this case, we also have .

3.1 Flow models in strong form

3.1.1 Stationary Stokes flow

Starting point is stationary Stokes flow on a manifold. Let u(x)∈C²(Γ) be the 3D velocity field on the surface Γ, p(x)∈C¹(Γ) a pressure field, and f_t(x) a tangential body force, eg, with unit . The governing field equations (in stress-divergence form37) to be fulfilled ∀x∈Γ are

(4)

(5)

(6)

Equation 4 expands to three momentum equations, Equation 5 is the incompressibility constraint and Equation 6 represents the tangential velocity constraint that restricts the velocities to the tangent space of Γ. Two different strain tensors are introduced

(7)

(8)

which are related to each other as ɛ^cov(u)=P·ɛ^dir(u)·P. The stress tensor is then defined as

where is the (constant) dynamic viscosity. It is easily shown that

Suppose there exists a boundary ∂Γ of the manifold that consists of two nonoverlapping parts, the Dirichlet boundary, ie, ∂Γ_D, and the Neumann boundary, ie, ∂Γ_N. The corresponding boundary conditions are given as

(9)

where the prescribed velocities and tractions are in the tangent space of Γ, ie, .

Note that, in general, there are no explicit boundary conditions needed for the pressure p. In cases where no Neumann boundary is present, ie, ∂Γ_N=∅ and ∂Γ_D=∂Γ, the pressure is defined up to a constant.37, 38 This includes compact manifolds where ∂Γ=∅. In such situations, the pressure may be prescribed at a given point on Γ or it is imposed by a constraint in the form of .

Vorticity on manifolds.

The vorticity ω is a physical quantity frequently computed in flow problems. In the context of manifolds, we shall define

(10)

Note that ω is colinear to the normal vector n_Γ; hence, P·ω=0. Therefore, it is useful to determine the signed magnitude of ω, ie, the scalar function

(11)

This scalar quantity may also be obtained using directional derivatives, ie, .

3.1.2 Stationary Navier-Stokes flow

For stationary Navier-Stokes flow, a nonlinear advection term is added to Equation 4, resulting into

(12)

where is the (constant) fluid density with unit and . It is quite common to express the body force in the form f_t(x)=ϱ·g_t(x), where g_t may consider gravity as for instance. The remaining Equations 5 and 6 and the boundary conditions 9 remain unchanged. The solution of the nonlinear governing equations can be obtained iteratively based on the Newton-Raphson method or other fixed-point iterations such as Picard iterations. Because the advection operator is not self-adjoint, well-known stability issues may arise for large Reynolds numbers in a numerical context.

3.1.3 Instationary Navier-Stokes flow

For instationary Navier-Stokes flow, the momentum equation 4 changes to

(13)

The functions representing the physical fields live in space (on Γ) and time, ie, in the time interval τ=[0,T]. Therefore, Equations 5-6, and 13 have to be solved in the space-time domain Γ×τ. Herein, we restrict ourselves to spatially fixed manifolds Γ.

The boundary conditions 9 also extend in time dimension; hence, there are prescribed velocities along ∂Γ_D×τ and tractions along ∂Γ_N×τ. Furthermore, an initial condition is needed

(14)

3.2 Flow models in weak form

The following trial and test function spaces are introduced:

(15)

(16)

(17)

(18)

As mentioned previously, if no Neumann boundary exists, ie, ∂Γ_N=∅, the pressure is defined up to a constant and one may replace by

3.2.1 Stationary Stokes flow

The weak form of the Stokes problem becomes the following. Given viscosity , body force f(x) in Γ, and traction on ∂Γ_N, find the velocity field , pressure field , and Lagrange multiplier field such that for all test functions , there holds in Γ

(19)

(20)

(21)

In order to obtain Equation 19, the divergence theorem 3 was applied to where the curvature term vanishes due to σ·n_Γ=0. Using the definition of the stress tensor, we get

The following relations are easily derived:

(22)

It is readily verified that solutions of the strong form also fulfill the weak form in Equations 19-21. This is obvious for Equations 20 and 21 due to Equations 5 and 6, respectively. For the momentum equations, it is noted that 19 is fufilled for −div_Γσ(u,p)+λ·n_Γ=f. Restricting this to the tangential space by multiplication with the projector P yields the strong form of the momentum equations 4 because P·n_Γ=0. It is thus also seen that the Lagrange multiplier field λ may be physically interpreted as a force in normal direction.

3.2.2 Stationary Navier-Stokes flow

The weak form of the stationary Navier-Stokes equations is similar to the aforementioned Stokes problem; however, Equation 19 is replaced by

where the added advection term is readily identified.

3.2.3 Instationary Navier-Stokes flow

The weak form of the instationary Navier-Stokes problem is as follows. Given density , viscosity , body force ϱ·g(x,t) in Γ×τ, traction on ∂Γ_N×τ, and initial condition u₀(x) on Γ at t=0, according to 14, find the velocity field , pressure field , and Lagrange multiplier field such that for all test functions , there holds in Γ×τ

(23)

(24)

(25)

3.3 Surface FEM for flows on manifolds

3.3.1 Surface meshes

Assume that a suitable surface mesh composed by higher-order triangular or quadrilateral Lagrange elements of order q may be generated with desired element sizes and all nodes on Γ. Well-known necessary requirements of meshes such as the shape regularity of the elements and bounds on inner angles are fulfilled. The shape of each (physical) element in the mesh results from a map of the corresponding reference element with n_q nodes

(26)

are classical Lagrangian shape functions of order q in reference coordinates and x_i∈Γ are the nodal coordinates. The resulting mesh is an approximation of the exact surface Γ. Clearly, is defined parametrically through the map 26 even if the original Γ was implicitly given, eg, by the zero isosurface of a level-set function. See other works39-41 for the automatic generation of higher-order meshes on zero isosurfaces. The discrete unit normal vector is

and is not smooth across element edges due to the C⁰-continuity of the surface mesh. The discrete tangent and conormal vectors and are easily obtained along the element edges on the boundary of . The definitions of the surface operators from Section 2.2 readily extend to the case of a discrete manifold and are not repeated here.

3.3.2 Surface FEM

We use higher-order surface FEM as detailed, eg, in the works of Demlow16 and Dziuk and Elliott18 for the discretization of the weak forms from Section 3.2. Finite element spaces of different orders are involved. As aforementioned, suitable surface meshes of order q may be generated defining approximations of Γ. Let there be a “geometry mesh” of order q=k_geom with the sole purpose to approximate the geometry of the manifold and define the element maps 26. In particular, this mesh is not used to imply a finite element space for the approximation of the weak forms.

Next, a finite element space of order k is generated on Γ^h for which it is assumed that there is a second mesh of order k. The two meshes feature the same element types and number of elements with identical coordinates at the corners; however, the total number of nodes differs due to the individual orders. It is emphasized that the coordinates of the nodes in the kth order mesh are, in fact, never needed and it is only the connectivity, which is required to set up the finite element space.

Associated to triangular or quadrilateral elements in the kth order mesh, there is a fixed set of local basis functions defined in a reference element with i=1,…,n_k and n_k being the number of nodes per element. Classical Lagrange basis functions with are used herein. Based on the map 26, which is completely determined by the geometry mesh, one may generate for all x∈Γ^h and tangential derivatives are determined based on Equation 2. This is only an isoparametric map when k=k_geom. Summing up the element contributions for nodes belonging to several elements, this generates a set of global C⁰-continuous basis functions in Γ^h with and being the number of nodes of the kth order surface mesh. Note that to generate the nodal basis , only the coordinates of the geometry mesh are needed; however, not from the kth order mesh. A general finite element space of order k is now defined by

Based on this, the following discrete trial and test function spaces are defined:

(27)

(28)

(29)

(30)

Although shape functions for the pressure and the Lagrange multiplier for enforcing the tangential velocity constraint may be discontinuous, we restrict ourselves to classical C⁰-continuous approximations. Note that individual orders k_u, k_p, and k_λ are associated to the approximations of velocities u^h, pressure p^h, and Lagrange multiplier field λ^h, respectively. Analogous to the continuous case, one may impose that the functions in have to fulfill if no Neumann boundary is present.

3.3.3 Stationary Stokes flow

The discrete weak form of the Stokes problem reads as follows. Given viscosity , body force f^h(x) in Γ^h, and traction on , find the velocity field , pressure field , and Lagrange multiplier field such that for all test functions , there holds in Γ^h

(31)

(32)

(33)

The usual element assembly yields a linear system of equations in the form

(34)

with =[u,v,w,p,λ]^T being the sought nodal values of the velocity components, pressure, and Lagrange multiplier. For the implementation, it is interesting to compare the system 34 with the system obtained for a classical 3D Stokes problem

(35)

Assume a function, which generates K_3D and G_3D based on 3D FE shape functions (including classical partial derivatives with respect to x), evaluated at given integration points in 3D. The same function may be used for generating K and G provided that (i) the integration points are restricted to Γ^h with proper weights; (ii) the classical partial derivatives in ∇ are replaced by the tangential derivatives as in ; and (iii) the contribution to K at the current integration point, ie, K(x_i), is projected as K(x_i)=P(x_i)·K_3D(x_i)·P(x_i), which is due to Equation 22. The same shall later hold for the advection matrix in the Navier-Stokes equations.

As expected in the context of the Lagrange multiplier method, the matrix in Equation 34 has a saddle-point structure and is typical for a mixed FEM. The well-known Babuška-Brezzi condition20-22 must be fulfilled to obtain useful solutions for all involved fields. This may be achieved by adjusting the orders of the approximation spaces for the different fields and is further detailed in the numerical results. It is noted that stabilization may be employed to circumvent the Babuška-Brezzi condition rather than to fulfill it, see, eg, in other works,22, 42, 43 which is, however, beyond the scope of this work.

3.3.4 Stationary Navier-Stokes flow

The discrete weak form of the stationary Navier-Stokes problem reads as follows. Given density , viscosity , body force ϱ·g^h(x) in Γ^h, and traction on , find the velocity field , pressure field , and Lagrange multiplier field such that for all test functions , there holds in Γ^h

The equations related to the different field equations were added up for brevity. The last row adds a stabilization term, which is needed to obtain stable solutions for flows at high Reynolds numbers.37, 38 In particular, the SUPG method is used for the stabilization. Different definitions of the stabilization parameter τ_SUPG are found30, 44, 45 and

is used herein with element-averaged velocity u_e, element length h_e, and Δt→∞ for the stationary case. When stabilization is not necessary because no oscillations occur, τ_SUPG=0. Note that, in the stabilization term, second-order derivatives appear (only in the element interiors). The definition of tangential second-order derivatives is given, eg, in the work of Delfour and Zolésio.35

Element assembly results in a nonlinear system of equations of the form

(36)

which is no longer symmetric (partly) due to the advection matrix . The distinguishing feature of K^⋆ and G^⋆(compared with K and G of the Stokes problem) are the added SUPG-stabilization terms. The issues related to mixed FEMs and the Babuška-Brezzi condition remain relevant.

3.3.5 Instationary Navier-Stokes flow

The discrete weak form of the instationary Navier-Stokes problem is the following. Given density , viscosity , body force ϱ·g^h(x,t) in Γ^h×τ, traction on , and initial condition on Γ^h at t=0 according to 14, find the velocity field , pressure field , and Lagrange multiplier field such that for all test functions , there holds in Γ^h×τ

This yields a system of nonlinear semidiscrete equations for t∈τ

with initial condition . This system may be advanced in time by using finite difference schemes, and the Crank-Nicolson method is employed herein.

4.1 Stokes flow on an axisymmetric surface

A test case is developed for which analytic solutions are available. An axisymmetric surface with height L=5 and radius

is generated as illustrated in Figure 2A. Let r₀=r(0) and . In parametrized form, one may also define Γ based on the map

The lower boundary at z=0 is the Dirichlet boundary ∂Γ_D, where the inflow in conormal direction of the manifold is prescribed as

with angle θ given by . The upper boundary at z=L is the outflow boundary where zero tractions are applied as Neumann boundary conditions. The density and viscosity are set to ϱ=1 and μ=0.01, respectively.

The mass flow on the lower boundary is

and due to mass conservation, the mass flow along the height follows as . As the flow field is expected to be axisymmetric for this test case and the tangential velocity constraint applies, one may compute the velocity components as

See Figure 3 for a graphical representation. It is noted that the mass flow Q(z), velocity magnitude , and the vertical velocity component w are only functions of z, ie, they do not vary in x- and y-directions.

Finite element approximations are carried out on various meshes composed by triangular or quadrilateral Lagrange elements of different orders. For the convergence studies, meshes with n_z={4,6,10,14,20,30,40,60} elements over the height are chosen; the number of elements in circumferential direction is . The meshes are perturbed, as illustrated in Figures 2B to 2D, to avoid perfectly axisymmetric meshes which, otherwise, could have improved the convergence rates for this special case.

The individual element orders used for the convergence studies are indicated by a four-tuple {k_geom,k_u,k_p,k_λ}. To be precise, this tuple summarizes the employed orders for the geometry, k_geom, the velocities, k_u, the pressure, k_p, and the Lagrange multiplier for enforcing the tangential velocity constraint, k_λ. For each tuple, meshes with different resolutions (given by n_z and n_θ) are considered and errors calculated, each time resulting in one curve in the convergence plots as indicated in the legends.

Systematic studies of different combinations of element orders showed that equal-order approximations for the velocity and pressure, ie, k_p=k_u, do not converge satisfactory (or at all), which is well known from the standard context of the incompressible Navier-Stokes equations in 2D and 3D due to the Babuška-Brezzi condition. For the studies outlined in this paper, we shall choose k_p=k_u−1, which is a popular choice for FEM approximations of classical incompressible flows and known as Taylor-Hood elements.46

For the first study, we use 2≤k_u≤5, k_p=k_u−1, and k_λ=k_u−1 which, later on, becomes the recommended standard setting. Convergence plots for ɛ_u and ɛ_p are given in Figure 4. The thick solid lines are for k_geom=k_u+1. It is noteworthy that, for quadrilateral elements, setting k_geom=k_u leads to almost identical results as seen from the thin dashed lines in Figures 4C and 4D. This does not necessarily hold for triangular elements; see Figures 4A and 4B where the convergence may drop by one order when setting k_geom=k_u rather than k_geom=k_u+1. This is later confirmed for the errors ɛ_mom and ɛ_cont in Figure 6. Therefore, we recommend to choose the geometry one order higher than k_u, which is done in the remainder of this work. Another reason is that the normal vector n is present in the governing equations and is computed based on the Jacobi matrix, ie, first derivatives of the element mappings of order k_geom are involved.

It is important to note in Figure 4 that the convergence rates in the pressure are optimal, ie, m_p=k_p+1; however, in the velocities one-order suboptimal, m_u=k_u. We have traced this back to the influence of the order k_λ of the Lagrange multiplier field. This is demonstrated in Figure 5 where Figure 5A shows the error ɛ_u and Figure 5B the condition number κ of the corresponding system of equations (obtained with MATLAB's condest function). As aforementioned, k_geom=k_u+1 and k_p=k_u−1. Figure 5 shows that setting k_λ=1 yields convergence rates m_u=2, independent of the other orders (black lines). Setting k_λ=k_u yields optimal convergence rates m_u=k_u+1 for the velocities (red lines); however, there is a dramatic influence on the conditioning, which scales with κ∼O(h⁻⁶) in this case rather than with O(h⁻²) for all choices where k_λ<k_u. Therefore, we set k_λ=k_u−1 in the following and accept the suboptimal convergence in the velocities.

Next, the error is observed in the strong form of the momentum and continuity equations, ie, ɛ_mom and ɛ_cont; see Equations 39 and 40. Results for k_p=k_λ=k_u−1 are depicted in Figure 6 for triangular and quadrilateral elements. Again, the thick lines refer to k_geom=k_u+1 and the thin dashed lines to k_geom=k_u. As aforementioned for the L₂-errors in the velocities and pressure, this makes a difference (of one order) for triangular elements, however, not for quadrilateral elements. When using k_geom=k_u+1 on the safe side, the convergence rate in ɛ_mom is m_mom=k_u−1 as expected due to the presence of second-order derivatives of u in the momentum equations. The expected convergence rate in ɛ_cont is m_cont=k_u due to the presence of first-order derivatives of u in the continuity equation.

4.2 Driven cavity flows on manifolds

The stationary Navier-Stokes model is considered in this example. Starting point is the driven cavity for the case of a flat 2D domain as depicted in Figure 7A. This case has well-documented reference solutions for a variety of Reynolds numbers.47 There, a flow inside a quadratic domain Ω_2D=(0,1)×(0,1) with no-slip boundary conditions on the left, right, and lower wall develops under a shear flow of u=1.0 and v=0.0 applied on the upper boundary until a stationary solution is reached. The Reynolds number is computed as Re=ϱ·u·L/μ.

Herein, the situation is extended to curved surfaces in 3D by deforming the flat 2D domain Ω_2D in z-direction using functions z(x,y). In particular, two different maps A and B are used

where α and β scale the height in z-direction; see Figures 7B and 7C for examples. The advantage is that, for α=0 and β=0, the flat situation is recovered and the reference solutions in the work of Ghia et al47 are relevant. We have confirmed that these solutions are recovered with great accuracy also for any rigid body transformation of Ω_2D into three dimensions. The density is chosen as ϱ=1 and two different viscosities of μ=0.01 and μ=0.001, leading to Reynolds numbers of Re=100 and Re=1000 for the flat case, respectively. Solutions for the velocity magnitude and pressure field for some example manifolds are displayed in Figure 8.

The meshes feature quadrilateral elements of different orders and are refined toward the boundaries to capture the resulting boundary layers. See Figure 9 for the meshes in Ω_2D, which are mapped to 3D according to maps A and B from above for various scaling coefficients α and β. The number of elements per dimension is n={10,20,30,50,70,100}. For the numerical studies, k_p=k_λ=k_u−1 and k_geom=k_u+1 is used as recommended previously.

Just as for the reference solutions in the work of Ghia et al,47 the results are presented as velocity profiles along the horizontal and vertical centerlines in Ω_2D. Figure 10 shows the profiles for the velocity component u along the vertical centerline and v along the horizontal centerline for the two maps with different scaling factors α and β, respectively. The crosses indicating the reference solution from the work of the aforementioned author47 are only relevant for the flat case where α=β=0. The results for the velocity component w along the two centerlines are given in Figure 11. These results have the quality of benchmark solutions and have been obtained with k_u=4 and 100 elements per dimensions. The convergence of other element orders and mesh resolutions toward these profiles have been confirmed, and a small selection is shown in Figure 12. Without stabilization, the typical oscillations are seen for this rather high Reynolds number for coarse meshes with low order. As no analytical solutions for the velocities and pressure are available, it is impossible to provide convergence results in ɛ_u and ɛ_p. Furthermore, the singular pressure in the upper left and right corners lead to singularities in the derivatives of other physical fields. Thus, it cannot be expected that (optimal) convergence in ɛ_mom and ɛ_cont is achieved.

4.3 Flows on zero-level sets

The next test case shows the potential to solve flows on zero-level sets with the proposed models. Stationary Stokes and Navier-Stokes flows are considered. The scalar function is based on the work of Dziuk and Elliott18 and defined as

The zero isosurface of ϕ implies the compact manifold of interest, ie, Γ={x:ϕ(x)=0}, and is depicted in Figure 13. In the first step, meshes with linear triangular elements are generated using distmesh.48 A scaling parameter h may be chosen, which defines an average element length. In the second step, higher-order elements are mapped to this linear surface mesh and their element nodes are “lifted”18 such that they are on the manifold Γ. Thereby, a higher-order accurate representation Γ^h is obtained.

As there are no boundaries present, an accelaration field in z-direction drives the flow. That is, on the right-hand side, , where g_z is determined by

and visualized in Figure 14A. It is virtually nonzero only for the left front “pillar” of the domain. The density is ϱ=1 and the viscosity is μ=0.05. For the case of stationary Navier-Stokes flow, the corresponding velocity magnitude, pressure fields, and vorticity ω^⋆ according to Equation 11 are seen in Figures 14B to 14D, respectively.

In the numerical studies, 2≤k_u≤5, k_p=k_λ=k_u−1, and k_geom=k_u+1 are used. As there is no analytical solution available, convergence results are only shown in ɛ_mom and ɛ_cont in Figure 15. Higher-order rates are clearly achieved. In order to make the solution more quantitative, the velocity profiles for w(x) in the horizontal xy-plane (at z=0) are shown in Figure 16. The four closed black lines represent the intersection of the plane with the vertical “pillars” of the zero isosurface. Figure 16A shows w(x) as a third dimension, and Figure 16B shows the same result where w(x) is plotted in normal direction of the plane-pillar intersections with a scaling factor of 0.4. A clear convergence to these profiles was observed when using meshes with different resolutions and orders.

4.4 Cylinder flows

As an example for the instationary Navier-Stokes equations, the following test case is based on a channel flow around a cylinder according to the work of Schäfer and Turek.49 The geometry is first described in 2D, labeled Ω_2D, and later on mapped to obtain curved surfaces in 3D. In 2D, the cylinder with a diameter of 0.1 is placed slightly unsymmetrically in y-direction of the channel in [0,2.20]×[0,0.41] (see Figure 17A). No-slip boundary conditions are applied on the upper and lower wall and on the cylinder surface. A quadratic velocity profile for u, with u_max=1.5, and v=0 is applied at the inflow on the left side of the domain. At the outflow, traction-free boundary conditions are used. The density and viscosity are prescribed as ϱ=1.0 and μ=0.001. This results in a Reynolds number of Re=ϱ·u_m·L/μ=100 when taking the cylinder diameter as a length scale L and the average inflow velocity u_m=1.0 at the inflow. At this Reynolds number, periodic flow patterns known as the Kármán vortex street are observed behind the cylinder. Reference solutions are given for the lift and drag coefficients c_L and c_D of the cylinder49 and the current implementation confirms these numbers for the flat case (ie, in 2D or when the flat 2D domain is transformed by a rigid body motion to 3D). The reference Strouhal number St=D/(u_mT), with the diameter D=0.1 of the cylinder, and the time T for 2 periods of the curve of c_D is given as 0.295≤St≤0.305, resulting in a frequency of about  Hz.

The 2D domain is mapped to three dimensions using two different maps. Assume that the coordinates of the 2D domain Ω_2D, as seen in Figure 17A, are given in coordinates (a,b). Map A, ie, , is defined as

For map B, we first define an intermediate mapping , applying some twist to the domain

with q(a)=−0.2/2.42·a²+0.44/2.42·a. This is further mapped by defined as

The resulting curved manifolds according to maps A and B are visualized in Figures 17B and 17C, respectively. Note that also the inflow velocities are mapped accordingly based on the Jacobians of the respective mappings to ensure that they are in the tangent space at ∂Γ_D.

The initial condition on the manifolds is u₀(x)=0. The observed time interval is τ=[0,6] and the inflow velocities are ramped by a cubic function in time

with t^⋆=0.96. That is, after t^⋆, the full velocity profile is active at the inflow. Figures 18 and 19 show the velocity magnitude, pressure field, and vorticity ω^⋆ at time t=6 for the two mappings. The expected vortex shedding can be clearly seen.

Two different meshes with 972 and 1920 elements each are used, which are refined at the no-slip boundaries to resolve the boundary layers. They are visualized for Ω_2D in Figure 20 and mapped to the manifolds accordingly. We use element orders of k_geom=4, k_u=3, k_p=2, and k_λ=2 in the numerical studies shown here. Higher orders achieved virtually indistinguishable results for the quantities shown below. It is also noted that the Crank-Nicolson method used for the time discretization is only second-order accurate. For the time discretization, n_step={150,300,600,1200,2400,4800} time steps are used. To make the results more quantitative, the stresses at the cylinder wall are summed up to obtain a force resultant in 3D. This is the equivalent of the lift and drag coefficients for the flat 2D case. Furthermore, the pressure difference between the front and back position of the cylinder (in Ω_2D, mapped to three dimensions) is computed, ie, Δp(t)=p_front(t)−p_back(t).

The results for map A are shown in Figure 21 for the different number of time steps. It can be seen that, after about 2 seconds, the expected vortex shedding is almost established. After 3 seconds, the resulting oscillations remain virtually unchanged. The time interval [5.2,6] is shown in more detail in Figures 21B and 21D for F(t) and Δp(t), respectively. The convergence with increasing number of time steps is clearly demonstrated. Figure 22 shows the results in the same style for map B; the same conclusions may be drawn. The spatial convergence is investigated in Figure 23 where it is found that the coarse and fine mesh employed here obtain very similar results for the chosen element orders. The frequency of the oscillations for map A is Hz and for map B is Hz; for the flat case, the frequency is Hz.