Abstract Classic Cartesian staggered mesh schemes have a number of attractive properties. They do not display spurious pressure modes and they have been shown to locally conserve, mass, momentum, kinetic energy, and circulation to machine precision. Recently, a number of generalizations of the staggered mesh approach have been proposed for unstructured (triangular or tetrahedral) meshes. These unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic Cartesian mesh method. This work addresses the momentum, kinetic energy, and circulation conservation properties of unstructured staggered mesh methods. It is shown that with certain choices of the velocity interpolation, unstructured staggered mesh discretizations of the divergence form of the Navier-Stokes equations can conserve kinetic energy and momentum both locally and globally. In addition, it is shown that unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations can conserve kinetic energy and circulation both locally and globally. The analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation. These novel conserving unstructured staggered mesh schemes have not been previously analyzed. It is shown that they are first-order accurate on nonuniform two-dimensional unstructured meshes and second-order accurate on uniform unstructured meshes. Numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented. INTRODUCTION Strictly speaking, a staggered mesh scheme is any numerical scheme where variables are located at different points within the mesh. Many possible staggering schemes are possible. However, in this work we are interested in generalizations of a particular staggering scheme that dates originally to the work of Harlow and Welch [1] . They describe a scheme for regular Cartesian meshes where the pressure is located at cell centers but the velocity is distributed on the cell faces with horizontal velocity components prescribed at vertical faces and vertical velocity components prescribed at horizontal faces. This particular staggering scheme has been found to be especially attractive for simulations of incompressible flow and is widely used for this class of flows. The important property for incompressible flows is the fact that this scheme does not display spurious pressure modes. There is no red-black uncoupling of the pressure unknowns or a need for "stabilization" terms that damp pressure and velocity fluctuations. INTRODUCTION Several properties beyond the ability to easily simulate incompressible flow make this method attractive for simulations of high Reynolds number flows. The method is typically very fast and uses minimal memory. In addition, it has been found that discretizations based on this staggering can conserve momentum, kinetic energy, and circulation [2] . Direct numerical simulations of turbulence have been successfully performed using the Cartesian staggered mesh discretization [3] [4] [5] , and more recently the scheme has become popular for Large Eddy Simulation (LES) in complex geometries [6] . Furthermore, it has been demonstrated that local conservation of kinetic energy is particularly critical for large eddy simulations of turbulence [7] , where it is observed that preserving the exact details of the small scale turbulent fluctuations is not nearly as important as preserving their overall kinetic energy and rate of dissipation. In two-dimensional turbulence, it is the enstrophy that becomes the important cascaded variable, and so it is likely in simulations of twodimensional (geophysical) turbulence that conservation of circulation (vorticity) will be an important property of the numerical method. INTRODUCTION Most numerical methods that are known to conserve kinetic energy or circulation, such as spectral methods or Cartesian staggered mesh schemes, require Cartesian meshes. If attention is restricted to methods that can be implemented on unstructured meshes and in complex geometries, the kinetic energy or circulation conservation properties are very restrictive. Mimetic discretizations on unstructured meshes which appear to be closely related to finite element methods can be constructed to globally conserve vorticity or enstropy or kinetic energy [8] . Galerkin finite element methods are known to be kinetic energy conserving (globally) on unstructured meshes. Unfortunately, experience with finite element methods for the LES simulation of turbulence indicates that they can also be expensive in that context [9] . In addition, finite element methods guarantee only global conservation of momentum and kinetic energy, the attractive local conservation properties often found with finite volume methods cannot be obtained. While finite volume methods do conserve mass, momentum, and total energy locally (usually by construction), in general, they do not conserve kinetic energy or circulation. In addition, standard finite volume methods are often subject to pressure instabilities and slow convergence at low Mach numbers. However, unstructured versions of the Harlow and Welch staggered mesh method hold significant potential for achieving local kinetic energy or vorticity conservation on unstructured meshes. INTRODUCTION The staggered mesh method of Harlow and Welch was generalized to unstructured (triangular) meshes independently by Hall et al. [10] and by Nicolaides [11] [12] [13] . The works of Nicolaides provide extensive mathematical analysis of the method. These "dual mesh" or "covolume" methods take explicit advantage of the fact that every unstructured tetrahedral or triangular mesh (a Delaunay mesh) has an orthogonal or dual mesh associated with it (a Voronoi tessellation). An example of an unstructured mesh and its dual are shown in Fig. 1 . The local mutual orthogonality of these meshes can be used to develop discretization operators that closely mimic their continuous counterparts. This allows a true inner product to be defined and important vector identities (such as ∇ · ∇ × (•) = 0) to be maintained in a discrete sense. Similar, "mimetic" schemes for distorted Cartesian meshes have recently been proposed by Bertagnolio and Daube [14] and Hyman and Shashkov [15, 16] . INTRODUCTION It should be noted that while unstructured staggered mesh methods have numerous attractive mathematical properties, high-order accuracy is currently not one of them. Like the original Harlow and Welch discretization, the methods tend to be first-order accurate on nonuniform meshes. The issue of creating higher order unstructured staggered mesh schemes is not pursued herein. However, Morinishi et al. [17] have developed higher order staggered mesh schemes for Cartesian staggered grids, and Hyman et al. [18] have reported progress in developing high-order staggered mesh schemes for distorted quadrilateral meshes. Calculations of fluid evolution using unstructured staggered mesh schemes are reported in Refs. [19] [20] [21] . INTRODUCTION This work will evaluate the conservation properties of two different unstructured staggered mesh schemes. For the sake of clarity and brevity the analysis will be restricted to two dimensions, but it will be clear throughout the analysis that there are no fundamental hurdles to applying the analysis to three-dimensional discretizations. Section 2 looks at unstructured staggered mesh discretizations of the rotational form of the Navier-Stokes equations. The rotational form is attractive because it maps well to the staggered mesh approach and is inexpensive to implement. The rotational form is shown to conserve circulation and kinetic energy locally and globally. Section 3 looks at unstructured staggered mesh discretizations of the divergence form of the Navier-Stokes equations. The method is shown to conserve momentum and kinetic energy locally and globally. Numerical confirmation of these conservation properties is presented in Section 4. The accuracy of these conserving staggered mesh discretizations is analyzed in Section 5, and a short discussion is presented in Section 6. ANALYSIS OF THE ROTATIONAL FORM Unstructured staggered mesh methods have discretization operators that are ideally suited to a representation of the Navier-Stokes equations that is based on the vorticity. The following form of the incompressible Navier-Stokes equations will be referred to as the rotational form of the equations, ANALYSIS OF THE ROTATIONAL FORM where u is the velocity vector, ω is the vorticity, ANALYSIS OF THE ROTATIONAL FORM u · u is the specific dynamic pressure, and ν is the kinematic viscosity. This equation assumes that viscosity is constant, but it is otherwise equivalent to other forms of the incompressible Navier-Stokes equations. Variable viscosity can be still be represented in rotational form but the extra term (involving second derivatives of viscosity) complicates the analysis unnecessarily. Despite the fact that this form of the Navier-Stokes equations is not common in textbooks it can be easily generalized to compressible flows and is a mathematically elegant way to view the equations. The convection term acts only perpendicularly to the velocity [u·(ω×u) = 0], and the body force is now explicitly decomposed into its dilatational and solenoidal parts. This particular form of the Navier-Stokes equations is of interest because it appears to be inherently suited to the staggered mesh discretization. The classic staggered mesh method of Harlow and Welch can be rearranged to look like a discretization of Eq. (1). Unstructured Discretization of the Rotational Form Staggered mesh schemes, either structured or unstructured, are concerned with the evolution of the normal velocity component, u, at the faces of the mesh cells. In two dimensions the staggered mesh discretization of the rotational form of the normal momentum equation at each cell face is Unstructured Discretization of the Rotational Form where W f , is the distance (or width) between neighboring cell circumcenters, A f is the face area, u is the normal velocity component as the face, ω n is the vorticity at the mesh nodes, and p Unstructured Discretization of the Rotational Form The orthogonality of the normal and tangential vectors can always be obtained if the cell positions are located at the cell circumcenters. The methods discussed in this manuscript assume that the cells have a circumcenter. This is true of meshes that consist of collections of triangles, rectangles, and symmetric trapezoids (and their three-dimensional counterparts), but is not true of a mesh composed of arbitrary quadrilaterals (or hexahedra). Extension of the staggered mesh method to arbitrary quadrilaterals is discussed in Ref. [15] . Arbitrary quadrilaterals force some of the discretization operators to be implicit, which complicates the analysis. While the following proofs require the presence of cell circumcenters, they do not require that the cell circumcenters be located within the cell itself or that the mesh be a Delaunay tessellation. Highly distorted grids can impact the accuracy of the unstructured staggered mesh method as discussed in [16] but do not affect its conservation properties. The test cases in Section 4 use meshes in which a significant fraction of the cell circumcenters lie outside the respective cell. In this notation, the velocity component normal to the cell face is u, and the tangential velocity component at the face is v. These local velocity components should not be confused with the x and y components of the velocity vector (which are given by u x and u y ). The convective term is actually calculated at the nodes and then averaged to the face. In two dimensions, the vorticity is assumed to point out of the two-dimensional plane. For convenience, we have represented the entire gradient term using a dynamic pressure, p d . Boundary faces are discretized in exactly the same manner, with the external cell (c2) located infinitesimally close to the boundary face. The time level has been dropped from all but the time derivative term for simplicity. However, the time levels are not necessarily arbitrary and some implications of various time level choices are discussed in the text. Unstructured Discretization of the Rotational Form If Eq. (2) is divided by W f A f , the discretization can be viewed as an unstructured finite difference approximation to the normal momentum equation. However, it can also be interpreted as a control volume approximation on a rectangular control volume with width W f and height A f . Note that the control volumes overlap and their total area will be exactly twice the area of the entire domain. This can be seen by the fact that each control volume rectangle is exactly twice as big as the two subtriangles associated with each face (see Fig. 2 ). The doubling of the total control volume area makes sense since the method solves for only half the velocity components. While the use of overlapping control volumes is perhaps troubling at first, it will be clear upon completion of the text that this is indeed a legitimate approach. If a regular Cartesian mesh is used it can be shown that this discretization is equivalent to the Harlow and Welch staggered mesh scheme. The accuracy of a similar staggered mesh scheme (with a different convection term) was analyzed by Nicolaides [11] and shown to be second-order accurate on uniform unstructured two-dimensional meshes. Unstructured Discretization of the Rotational Form Analysis of the method is made somewhat easier if the discrete equations are written in operator form, Unstructured Discretization of the Rotational Form where AVG is the averaging operator, GRAD is the gradient operator, and CURL is the curl operator. These operators are nonsquare, sparse matrices. The nonzero entries in the difference operators GRAD and CURL are either plus or minus one. The gradient and curl operators have a similar function to their continuous counterparts but differ slightly in that these matrices have been stripped of their geometric information such as heights and widths. It can be shown [22] that (CURL) T GRAD(•) = 0, which is the analog of ∇ × ∇(•) = 0. We will not use this fact in what follows but it highlights an important point: the staggered mesh operators have many properties that mimic their continuous counterparts [23] . What will be used in the conservation proofs will be discrete integration by parts analogs for these discrete operators. Discrete integration by parts will allow us to develop proofs of conservation that proceed very much like their continuous counterparts. Conservation of Kinetic Energy: Rotational Form For the continuous Navier-Stokes equations, kinetic energy conservation is derived by taking a dot product of the momentum equation with the velocity vector and using integration by parts. If we continue to consider an incompressible fluid then kinetic energy conservation is given by the equation, Conservation of Kinetic Energy: Rotational Form where 1 2 u · u is the specific kinetic energy of the fluid. Strictly speaking, conservation applies only in the inviscid limit of zero viscosity. In the presence of viscosity the last term is a negative semidefinite sink term that causes the total kinetic energy to decay monotonically in the absence of external forces. The discrete system will actually mimic this more general notion of kinetic energy conservation. It will satisfy a discrete equation analogous to the continuous equation above, where total kinetic energy decreases due to a single dissipation term which is equal to the product of the discrete enstrophy and viscosity. Conservation of Kinetic Energy: Rotational Form The primary difficulty of analyzing the conservation properties of staggered mesh methods is the fact that only the normal component of the velocity vector at the mesh faces is discretized. The choice of velocity interpolation is then intimately associated with how velocity-dependent quantities like momentum and kinetic energy are defined. During the course of the proof we must also determine how best these quantities should be defined. Conservation of Kinetic Energy: Rotational Form To show discrete kinetic energy conservation we multiply each equation for the normal velocity at the face by the half-time normal velocity, u n+1/2 , at the face and then sum over the faces. Note that in the discrete derivation there is no dot product, only the normal momentum equation and normal velocity are involved. Discrete conservation of kinetic energy will be derived by showing that a discrete form of integration by parts can be applied to each term of the normal momentum equation. Then it will be shown that this portion of the kinetic energy (that due to 1 2 u 2 summed over all cell faces) is an approximation for the full kinetic energy within each mesh cell. Conservation of Kinetic Energy: Rotational Form If the summation is performed over all the faces of a single control volume then the following proof is a statement of local kinetic energy conservation. It states that the change in kinetic energy within the control volume is only a result of fluxes through the control volume surfaces and viscous dissipation. If the summation is performed over all the faces in a mesh (including the boundary faces) then the interior fluxes cancel out and the resulting proof is a statement of global conservation. Global conservation states that the change in total kientic energy is due only to fluxes through the boundary of the domain and the total viscous dissipation inside the domain. Conservation of Kinetic Energy: Rotational Form Discrete conservation starts with the following equation: Conservation of Kinetic Energy: Rotational Form The goal will be to move u n+1/2 within each of the operators (including the time derivative) in order to derive a discrete analog of Eq. (4 Conservation of Kinetic Energy: Rotational Form Remember that the width W f is the distance between the neighboring two cell circumcenters (Fig. 2) . On boundary faces W f is the distance from the interior cell circumcenter to the face midpoint. The width can be decomposed into two parts, each representing the distance from the face midpoint to one of the cell circumcenters; Conservation of Kinetic Energy: Rotational Form f . On boundary faces the second part is zero. With this notation, the sum over faces can be recast as summation over the cells, Conservation of Kinetic Energy: Rotational Form where the term in square brackets can be identified as the change in discrete kinetic energy within the cell. The discrete kinetic energy in a cell is therefore defined by the expression Conservation of Kinetic Energy: Rotational Form where V c is the volume of the cell. Note that the summation involves only the "kinetic energy" associated with the normal velocity components. Also note that the summation is effectively a volume weighted average but that the weights add up to 2, so the weights account for the fact that only part of the total kinetic energy at each face is being averaged. It is shown in Section 5.2 that the average described above is a first-order accurate approximation for the kinetic energy in an arbitrary two-dimensional cell. Conservation of Kinetic Energy: Rotational Form With the discrete kinetic energy defined in this way the time derivative term for the discrete kinetic energy equation can be written as Conservation of Kinetic Energy: Rotational Form This equation is valid for a collection of cells or an individual mesh cell. Note that for unsteady flows kinetic energy conservation requires using a standard time derivative approximation (u n+1 − u n )/ t and the midpoint rule for the half-time velocity Conservation of Kinetic Energy: Rotational Form Setting u n+1/2 = u n in the derivation or using a second-order backward difference for the time derivative results in a first-order, but not necessarily positive, definite approximation for the kinetic energy at the next time level. Convective term. Here we will attempt to construct a discrete version of the convective term, (ω × u) · n, which gives no contribution to the discrete kinetic energy equation since this term does not contribute in the continuous kinetic energy equation. Convective term. The convective term in the kinetic energy is given by Convective term. where ω n is the vorticity at the node. The component of velocity that is used is the component of velocity oriented 90 Convective term. • counterclockwise to the face normal vector. This is precisely written as Convective term. where z is the unit vector pointing out of the two-dimensional plane. The summation above can be recast as a summation over mesh nodes: Convective term. The cross product term in large brackets is the term of critical interest. If it is desired that this term be identically zero for arbitrary flows then the term in square brackets must be proportional to the velocity at the nodes (v n ). This can be achieved by defining the node velocity as Convective term. where A n is the area associated with each node (see Fig. 1 ). Section 5.3 shows that this is a first-order approximation for the velocity on uniform meshes. However, on nonuniform meshes it is found that this is only a zeroth-order approximation for the velocity at the nodes. So while this discretization conserves kinetic energy, it has sacrificed convergence to the Navier-Stokes equations. While the error is shown to be small in Section 5.5, it is nonetheless disconcerting. A modified discretization of the convective term is presented below which leads to a first-order accurate velocity at the nodes. Modified convective term. The analysis in Section 5.3 which shows that the velocity defined by Eq. (13) is zeroth-order accurate for arbitrary meshes also shows how to improve the accuracy to first order. The first-order expression for the node velocity is similar to Eq. (13). Modified convective term. where x * f is the face position midway between the two neighboring cell circumcenters, andû is the component of velocity normal to the face and oriented in a counterclockwise direction with respect to the node. For uniform meshes, x * Modified convective term. A f n f and Eq. (13) equals Eq. (14) . Working backward from Eq. (14) , this implies that we would like Eq. (12) to have the form Modified convective term. Recasting this as a summation over faces gives Modified convective term. which simplifies to Modified convective term. This implies that the rotational form of the convection term (ω × u) · n should be discretized as Modified convective term. rather than using Eq. (11) . Note that Eq. (10) still holds but we now have a new definition for the tangential velocity, v n . Section 5.3 shows that Eq. (18) is still a first-order approximation for the convection term. Note that conservation of kinetic energy requires that the velocity in the convection term should be evaluated at the half time level, but there are no restrictions on the time level of the vorticity. Gradient term. This Section will show how the GRAD operator satisfies a discrete version of the chain rule analogous to u · ∇φ = ∇ · (uφ) − φ(∇ · u). The gradient term in the discrete energy equation actually contains two terms, a pressure contribution and a kinetic energy contribution. The kinetic energy portion will eventually become the convection of kinetic energy. The pressure portion becomes the traditional pressure work term. If we ignore the initial minus sign for now, the gradient term in the kinetic energy can be rearranged as follows: Gradient term. Note that the normal velocity component has an orientation (from C1 to C2), and the dynamic pressure at the cell faces p d f has been introduced. This latter step is not necessary, and the proof can proceed by simply remembering that at boundary faces of the domain the outer cell (C2) position is the same as the face position (so p Gradient term. . However, the current construction makes the chain rule analogy mentioned above more obvious. The summation over faces can now be rewritten as a summation over cells, Gradient term. where the normal velocityû is defined to be the normal component of velocity that points outward relative to the cell in question (rather than from C1 to C2). The term in square brackets is the discrete version of the velocity divergence. This derivation shows that in operator notation we can write Gradient term. where DIV = GRAD T is the discrete divergence operator. The summation over cell faces is required in the discrete version, because the discrete operators do not operate on vector quantities, the vector nature of the continuous analog is implicitly obtained by summation over the components oriented in different directions. Gradient term. If the solution method satisfies the discrete continuity equation, DIV(u A f ) = 0, then the gradient term in the kinetic energy equation becomes Gradient term. which is a pressure work term and a kinetic energy convection term. Note that when the summation occurs over many cells the contributions from faces with two cells (interior faces) cancel out because the outward pointing velocityû n+1/2 is equal and opposite for the two cells. Only boundary faces (with one cell contribution) survive, so this summation can be further simplified to Gradient term. whereû n+1/2 always points out of the domain. This indicates that the gradient term neither creates nor destroys discrete kinetic energy; it only moves it around. The same is true in the continuous case. Viscous term. This Section will show that the CURL operator also satisfies a discrete version of the chain rule for differentiation. It is analogous to Viscous term. The viscous term in the kinetic energy equation is Viscous term. This expression can be converted into a summation over mesh nodes, Viscous term. whereû is the velocity component normal to the face and pointing counterclockwise with respect to the node in question. If a node is in the interior of the domain, then the summation over node faces is a discrete approximation for the vorticity at the node (times the node area). Viscous term. On boundary nodes, the discrete vorticity must be completed by integrating the velocity around the boundary faces connected to the boundary node. Define the vorticity at a node using a discrete version of Stokes theorem, Viscous term. where t is the tangential vector on the boundary faces (oriented counterclockwise). The last term is necessary only for boundary nodes, and the normal vector is assumed to point outward at boundary faces. Then the viscous term becomes Viscous term. The second term can be converted to a summation over boundary faces: Viscous term. The first term is an approximation for dissipation in the domain. The second term is an approximation for ∇ · (νu × ω) in the domain. The approximation for u × ω at the faces is the same as that used for the convective term earlier. These terms correspond directly to what is found in the viscous terms of the continuous kinetic energy equation (Eq. (4)). Viscous term. Note that the dissipation is strictly negative definite only if the vorticity in the diffusion term is evaluated implicitly with ω n = ω n+1/2 n . This is often advisable for stability reasons anyway. Summary. It has been shown that in two dimensions an unstructured staggered mesh discretization of the rotational form of the Navier-Stokes equations can satisfy the following equation: Summary. where the discrete kinetic energy is given by Summary. This is a discrete analog of the continuous kinetic energy transport equation. It is conservative in the absence of viscosity and purely dissipative in the presence of viscosity, with no artificial dissipation. One subtle assumption in the derivation was in the form of the convective term itself. The convective term was evaluated by constructing ω × u at the nodes and averaging that result to the faces. Other possibilities, such as averaging the vorticity and velocity to the faces first and then taking the cross product, have not been evaluated. Summary. It is interesting to note that the definition of the discrete vorticity defines a new operator, ROT (The notation ROT is borrowed from Ref. [8] , which uses it in a similar context.): Summary. It turns out that ROT = CURL T , so the rotational form of the viscous term in the momentum equation is a symmetric positive semidefinite operator. This makes it relatively easy to invert via iterative methods and suitable for implicit solution. Conservation of Vorticity: Rotational Form For the continuous Navier-Stokes equations, vorticity conservation is derived by taking the curl of the momentum equation. If we continue to consider an incompressible fluid then vorticity evolution is given by the equation Conservation of Vorticity: Rotational Form which can also be written as Conservation of Vorticity: Rotational Form where ω is the vorticity. In two dimensions the first term on the right-hand side of Eq. (33) is zero and vorticity is a conserved variable. In an inviscid incompressible two-dimensional flow all the moments of vorticity are also conserved quantities. In three dimensions, Eq. (32) is the more useful viewpoint and it indicates that in three dimensions the circulation (integral of velocity around a closed loop) is a conserved quantity. However, we continue to focus on the two-dimensional case in this section. Conservation of Vorticity: Rotational Form To show discrete vorticity conservation we use a discrete curl operation, specifically the ROT operation defined previously. This is equivalent to a counterclockwise line integral around the faces of the dual control volume surrounding a node. This is specifically performed by dividing the momentum equation by the face area (so it becomes an approximation for the momentum equation integrated along the face length), multiplying by −1 if the face normal points clockwise with respect to the node in question, and finally summing over all the faces touching a specific node. The result is a proof of local vorticity conservation. Global conservation is then shown by proving that the fluxes at neighboring node control volumes cancel out everywhere in the interior of the domain, leaving only boundary contributions. Nicolaides shows an alternative method of proving vorticity conservation in Ref. [22] . Discrete conservation of vorticity at a single node starts with the following equation, Conservation of Vorticity: Rotational Form where the normal vector at each face has been chosen to point in a direction counterclockwise with respect to the node in question. The goal is to recast this equation into a discrete analog of the continuous vorticity transport equation given by Eq. (33). For the time being it will be assumed that the node or nodes in question are in the interior of the domain. Boundary nodes will be treated in Section 2.3.2. Local vorticity conservation. If the node is an interior node then the sum over all node faces represents a closed line integral around the node, and the time derivative becomes an approximation for the change in vorticity at the node times the node area: Local vorticity conservation. The convection term can be expanded as follows, Local vorticity conservation. where node n0 is the node over which summation is occurring and the face normal vector is assumed to point counterclockwise with respect to node n0. The node ni is a neighboring node to n0 connected by the face in question. This can be rewritten in the simpler form, Local vorticity conservation. where v fn is the velocity along the line pointing from the face to the node and outward. Each face contribution represents an approximation for the flux of vorticity into the node dual control volume (Voronoi cell). For uniform meshes, the velocity component is exactly perpendicular to the dual control volume faces. For nonuniform meshes the velocity is not exactly perpendicular, but the two tangential velocity components nearly cancel and the approximation is still a first-order approximation for the normal flux at the face. The curl of a gradient is zero for continuous differential operators. The same is also true of the discrete unstructured staggered mesh operators, so the discrete curl operation also eliminates the pressure term in this discrete vorticity equation: Local vorticity conservation. The cancellation occurs only if the faces completely surround the node, so this is not true for boundary nodes unless boundary conditions are introduced. Finally, the viscous term becomes Local vorticity conservation. which is a symmetric, negative semidefinite, conservation diffusion term. The final equation becomes Local vorticity conservation. where the convection velocity at each face is given by v nf = (x * f − x ni ) · v ni 2/A f . Note that Eq. (40) can be interpreted as a control volume discretization of the vorticity equation integrated over the dual mesh control volumes (Voronoi cells) surrounding each node. Global vorticity conservation. Global conservation is a direct consequence of the fact that the convective and diffusive vorticity fluxes at interior faces are equal and opposite for the two nodes touching each face. Consequently, the contributions from all interior faces cancel out. Global vorticity conservation. However, to rigorously account for all the vorticity in the domain it is necessary to include the vorticity contributions from boundary nodes and their associated dual control volumes. In order to account correctly for the vorticity at boundary nodes, the dual control volumes must be closed by connecting the centers of the two boundary faces touching each boundary node. If boundary conditions are used to define appropriate approximations to the momentum equation along this new dual control volume face, then the analysis of Section 2.3.1 above remains valid. Global vorticity conservation. When global conservation is analyzed by summing over the area weighted vorticity from all nodes in the domain, including boundary nodes, the fluxes from the new dual control volume faces at the boundaries do not cancel. The result is that the net change in total vorticity is due to the vorticity fluxes through these dual control volume faces at the boundaries. ANALYSIS OF THE DIVERGENCE FORM The next discretization that will be considered is based on the well known divergence form of the Navier-Stokes equations: ANALYSIS OF THE DIVERGENCE FORM Discretizations based on the divergence form of the equations are of interest because they are expected to be able to discretely conserve linear momentum. Note that conservation of momentum is by no means guaranteed by using this form, because the staggered mesh methods update only the normal velocity components at cell faces; tangential velocity components are interpolated. The primary disadvantage of the divergence form is that in the context of unstructured staggered mesh methods the divergence form requires more computational and memory overhead. Unstructured Discretization of the Divergence Form At interior faces the orientation of the normal is chosen to point from cell C1 to cell C2, at boundary faces the normal vector is assumed to point out of the domain. The discrete equation for the evolution of the normal velocity component is given by Unstructured Discretization of the Divergence Form where c c = cell faces ν(∇u +∇u T ) ·n f A f is a conservative discretization of the diffusion term evaluated in each cell, V c is the volume of each cell (which is really an area times unit depth), and W f C is the distance between the face circumcenter and the cell circumcenter. Remember that u = u · n f is the normal velocity at each cell face andû is the normal velocity component that points out of the cell. This discretization continues to assume that the mesh cells have a unique circumcenter, so that the face normal continues to point directly away from one cell center and directly toward another cell center. The following analysis will indicate that specific interpolations for the face velocity, u f , and velocity gradient at the cell faces, ∇u, are necessary in order to conserve discrete kinetic energy. Conservation of Kinetic Energy: Divergence Form The proof of discrete kinetic energy conservation of the divergence form begins just like the proof for the rotational form. Each equation for the normal velocity at the face is multiplied by the half-time normal velocity, u n+1/2 , and then summed over the faces. Discrete conservation of kinetic energy is derived by showing that a discrete form of integration by parts can be applied to each term of the normal momentum equation. Then it is shown that this portion of the kinetic energy (that due to the normal velocity summed over all cell faces) is an approximation for the full kinetic energy within each mesh cell. Conservation of Kinetic Energy: Divergence Form Discrete kinetic energy conservation of the divergence form starts with the following equation: Conservation of Kinetic Energy: Divergence Form The time derivative remains the same as in the standard rotational discretization (Section 2.2.1) and will not be reevaluated. The pressure gradient term has also been analyzed in the context of the standard rotational discretization (Section 2.2.4); the only difference is that in this context the pressure gradient term contains only the pressure, not the dynamic pressure. The following analysis will therefore focus on the convection and diffusion terms. Convection term. The convection term can be rewritten as a summation over cells, Convection term. where Convection term. A f is an approximation for the velocity vector in the cells. Section 5.4 proves that this is a first-order approximation for the velocity vector. Convection term. Expanding the convection vector, c, it is found that Eq. (44) becomes Convection term. If the face velocity u f at interior faces is required to be a simple average of u * c from the neighboring cells then Convection term. This can be converted back into a summation over cells which then disappears due to the continuity constraint, Convection term. where u * E = 2u f −u * c1 is an extrapolated velocity at the boundary faces, resulting in a secondorder approximation for the kinetic energy at the boundary faces. The result indicates that kinetic energy fluxes cancel out on the interior faces, and the only net convection of kinetic energy occurs through the boundaries. Diffusive term. The diffusive term can also be written as a summation over cells: Diffusive term. For simplicity we assume constant viscosity and incompressible flow. Then expanding the diffusion vector, d, it is found that this expression becomes, Diffusive term. where u * c2 equals the face velocity on boundary faces. This is equivalent to Diffusive term. The first term is a discrete approximation for the dissipation rate. If the velocity gradient at the faces is evaluated using u * c then the dissipation is a negative semidefinite quantity. The second term is the viscous work term at the boundaries of the domain. Summary. It has been shown that for constant viscosity, the two-dimensional divergence form of the unstructured staggered mesh scheme satisfies the following discrete kinetic energy equation, Summary. where the discrete kinetic energy is given by Summary. and the discrete dissipation is given by Summary. The construction of the discrete kinetic energy evolution equation required that cell velocity and face velocity interpolations were based on Summary. which is a first-order accurate approximation of the cell velocity for arbitrary two-dimensional meshes. Conservation of Momentum: Divergence Form For standard control volume discretizations, momentum conservation is a straightforward consequence of writing the equations in divergence form. However, a proof of conservation of momentum is far less obvious for staggered mesh numerical schemes on unstructured meshes. The inherent difficulty is due to the fact that the velocity vector is not a uniquely defined quantity in such methods. Part of the proof of momentum conservation will be to derive how the velocity vector should best be defined. Conservation of Momentum: Divergence Form To show discrete momentum conservation we multiply each equation for the normal velocity at the face by the face normal vector, and sum over the faces. Initially, this appears to be an incomplete procedure since only a portion of the momentum at each face is being analyzed. However, it will be shown that this can be reinterpreted as evolution equation for the velocity vector in each cell, and this evolution equation is conservative. Conservation of Momentum: Divergence Form If the summation is performed over all the faces of a single control volume then the following proof is a statement of local momentum conservation. It states that the change in momentum within the control volume is only a result of fluxes through the control volume surfaces. If the summation is performed over all the faces in a mesh (including the boundary faces) then the interior fluxes cancel out and the resulting proof is a statement of global conservation. Global conservation states that the change in momentum is due only to fluxes through the boundary of the domain. Conservation of Momentum: Divergence Form Discrete momentum conservation of the divergence form starts with the following equation: Conservation of Momentum: Divergence Form The goal will be to recast this equation as an equation for the cell velocity vector. Critical to this transformation is the definition of the cell velocity vector in terms of the face normal velocity components. Each term in this equation is evaluated below. Time derivative. Remember that the width W f is the distance between the neighboring two cell circumcenters (Fig. 2) . On boundary faces, W f is the distance from the interior cell circumcenter to the face circumcenter. The width can be decomposed into two parts, each representing the distance from the face circumcenter to one of the cell circum- Time derivative. f . On boundary faces the second part is zero. With this notation, the sum over faces can be recast as summation over the cells, Time derivative. where the term in square brackets can be identified as the change in the velocity vector within the cell. The velocity vector in a cell is therefore defined by the first-order approximation, Time derivative. where V c is the volume of the cell. With the cell velocity vector defined in this way, the time derivative term can be written as Time derivative. This equation is valid for a collection of cells or an individual mesh cell. Convection term. The convection term can be rewritten as a summation over cells: Convection term. The term in brackets is equal to the identity tensor I, times the cell volume. This is a known geometric identity, but the user can prove it using the basic techniques described in Section 5. We can now expand the convection term using its definition, = cells cell faces Convection term. There are two contributions from each interior face, and they exactly cancel out, sinceû is equal and opposite for the two contributions. The remaining term is the flux of momentum into the domain, across the boundary faces. Pressure gradient term. The gradient term in the kinetic energy can be rearranged as wheren f is the normal vector pointing out of the cell, or in the case of boundary faces-out of the domain. Diffusion term. The diffusion term behaves very similarly to convection, Diffusion term. where the bracketed term can be replaced by the identity matrix for two-dimensional meshes. Expanding the diffusion vector, d, it is found that cells cell faces Diffusion term. Again, there are two contributions from each interior face, and they exactly cancel out, sincen f is equal and opposite for the two contributions, and all other quantities are identical. The remaining term is the viscous flux of momentum into the domain, across the boundary faces. Diffusion term. Note that the rotational form of the viscous term also conserves momentum. The rotational form is Diffusion term. On interior nodes the last summation is zero. On boundary nodes the summation over node faces is simply the difference between the neighboring two boundary face circumcenters, so we obtain Diffusion term. The summation over face nodes is a discrete approximation for the normal component of the diffusive term integrated over each boundary face. The entire summation is a firstorder approximation for the total diffusion of velocity within the domain. While this is not the classic divergence form of the diffusion term this representation is equally valid and conservative. Diffusion term. 3.3.5. Summary. It has been shown that the divergence form of the unstructured staggered mesh scheme satisfies the following first-order conservative equation for the evolution of the cell velocities (or momentum), Diffusion term. where the cell velocity is given by the first-order approximation Diffusion term. This indicates that the unstructured staggered mesh discretization of the divergence form of the Navier-Stokes equations is fully equivalent to a control volume method for the velocity vector located at cell centers. NUMERICAL TESTS OF CONSERVATION In order to test the conservation properties of these schemes a problem was chosen that has zero mass flux at the boundaries, but is inherently unsteady. The problem involves a roughly circular patch of constant vorticity (magnitude of 2 s −1 ) located in the bottom left quadrant of a square domain. The domain boundaries are slip walls. The vortex patch is 10 cm in diameter and the square domain is 100 cm. The vortex rotates counterclockwise with a maximum speed of 6 cm/s, and its center moves counterclockwise around the domain at a speed of roughly 0.5 cm/s. A total of 6200 triangles are used to spatially discretize the domain. The initial streamlines are shown in Fig. 3a . The mesh and the initial vortex patch are shown in Fig. 3b. Conservation of Kinetic Energy The discrete kinetic energy within each mesh cell was calculated at every time step using Eq. (8) . The average discrete kinetic energy was then evaluated by calculating the volume weighted sum over all the cell kinetic energies, KE average = 1 V domain cells V c K c . Since there is no flow across the domain boundaries, this quantity should be constant in the absence of viscosity, and monotonically decreasing in the presence of viscosity with a decay rate proportional to the viscosity times the enstrophy. Conservation of Kinetic Energy In numerical tests of the vortex motion in the absence of viscosity, the average discrete kinetic remained constant to six significant digits after 200 time steps (10 s). This is about as constant as can be expected given the tolerances prescribed for the iterative solver. This level of accuracy was achieved for both the divergence and rotational discretizations. Figure 4a shows the change in the average discrete kinetic energy as a function of time when viscosity is present (0.1 and 0.001 cm 2 /s). Both the rotational (dashed lines) and the divergence forms (solid lines) are shown. The lower two curves are the higher viscosity case. In both cases, the divergence form tends to dissipate more kinetic energy than the rotational form in the early stages of the evolution. It is hypothesized that this is because the divergence form contains two extra averaging operations as compared to the rotational form. Since the vorticity is initially discontinuous, these extra averaging operations introduce artificial dissipation in the early stages of the vortex evolution. After roughly one diffusion time scale the vorticity becomes smoother and the two methods dissipate at nearly equal rates, as seen for the higher viscosity case. The dissipation rate (time derivative of kinetic energy) for the rotational discretization is examined in more detail in Fig. 4b . This figure shows the previous two viscosity values and the actual derivative of kinetic energy (lines) compared to the average discrete dissipation rate theoretically determined in the text ( 1 V domain nodes νω 2 n A n ) which is represented by symbols. The match between symbols and lines indicates that the theoretical analysis of Section 2.2 is well founded. Conservation of Vorticity The discrete vorticity within each mesh cell was calculated at every time step using Eq. (26). The average discrete vorticity was then evaluated by calculating the area weighted sum over all the individual node vorticity values, average = 1 A domain nodes A n ω n . Since there is no flow across the domain boundaries, and the diffusive vorticity flux is zero at the domain boundaries, this quantity should be constant, even in the presence of viscosity. figures after the solution has evolved for 10 s (200 time steps). The divergence form (solid lines) does not conserve vorticity. Conservation of Vorticity Theoretically, the total enstrophy (vorticity squared) is conserved in the absence of viscosity. However, this conservation property is not captured by either the rotational or the divergence discretization methods. Conservation of Momentum The discrete velocity (momentum) within each mesh cell was calculated at every time step using Eq. (55). The average discrete velocity was then evaluated by calculating the volume weighted sum over all the individual cell velocity values, Conservation of Momentum Since there is no flow across the domain boundaries, and because we are using slip walls, this quantity should remain constant even in the presence of viscosity. Conservation of Momentum It was found that both discretizations conserved the average x and y momentum to machine precision. The global conservation of momentum of the rotational discretization was not shown analytically in the text, but was determined to be a result of the fact that the streamfunction on the boundaries does not change with time. Conservation of Momentum A more difficult test of conservation was also performed by superimposing a uniform x velocity (5 cm/s) on the previous test problem and allowing the left boundary to be an inflow condition and the right boundary to be an outflow condition. This situation has a time varying streamfunction at the outflow. The divergence discretization continued to conserve x momentum to machine precision (y momentum is not constant for this problem). ACCURACY OF STAGGERED MESH METHODS In this section a new approach to determining the order of accuracy of unstructured staggered mesh discretizations is presented. This method as not based on Taylor series expansions which become unwieldy for two-dimensional unstructured meshes. Nor is it based on showing that the solution lies in a function space of piecewise polynomials of a certain order as is customary in finite element methods. ACCURACY OF STAGGERED MESH METHODS Instead it will be shown that discrete versions of Gauss'
{'paper_id': '18323632', 'title': 'Conservation Properties of Unstructured Staggered Mesh Schemes', 's2_url': 'https://api.semanticscholar.org/CorpusID:18323632', 'source': 'S2ORC'}