Post Mixing Optimization and Solutions

CFD Background

This is a continuation of the Process Intensifier - Optimization with CFD: Part 1 paper.

Fundamental Formulation

The mathematical foundations of the commercially available AcuSolve CFD code are principally the Galerkin/Least-Squares (GLS) finite element method, along with the development and implementation of consistent operators to guarantee conservation of all the primary variables. This formulation has the characteristic that it has been RIGOROUSLY PROVEN MATHEMATICALLY to be STABLE and CONVERGENT to the differential equations of the original physics, as a function of the mesh discretization. This approach maintains stability without sacrificing accuracy (Hughes et al 1989, Johnson and Szepessy 1985, Johnson and Szepessy 1986). This statement can not be made by the other major commercial finite volume and other finite element CFD codes, since up to this point in time, comparable mathematical proofs do not exist.

This characteristic alone engenders predictive confidence that the solutions are representative of the physics and hence accurate within the limits of the discretization. This provides the ability to accurately solve a wide range of model and fluid rheology variations.

This factor is one of several that contributes to the robustness / stability, and solvability of general CFD flows, which also permits the solution of a wide range of flow conditions from creeping Stoke's flows to very high Reynolds numbers. (Data has been shown for this (Zalc et al 2002) for SMX type mixing flows over a range of Reynolds numbers - from the order of 10E-6 to 10E+2, with variation from experimental data of less than 2% with no arbitrary tuning). Further articles and publications have demonstrated the agreement between physics (by experiment and analytical solutions) and AcuSolve results (see

The formulation includes discontinuity-capturing operators, along with other non-linear effects, to accurately handle the effects of such subjects as flows around sharp corners.

The fluid mechanics is implemented in a general fashion permitting a wide range of rheology behaviors, from Newtonian fluids through arbitrarily tabulated viscosity relationships.


From a mathematics / physics perspective, AcuSolve has implemented the linear algebra algorithms with fully coupled velocity and pressure solutions as well as both segregated and fully coupled energy equations depending on the specific needs. This, along with the high accuracy formulation, provides high quality tangent matrices for the Newton-class nonlinear iterative solver.

From a computational perspective, AcuSolve is implemented using high performance parallel linear algebra, fully adaptable to shared, distributed, and hybrid memory parallel computing architectures.

Turbulence Models

AcuSolve has implemented a number of turbulence models, including Spalart-Allmaras, Large Eddy Simulation (LES) classical Smagorinsky model, Detached Eddy Simulation (DES) (which is a hybrid Spalart-Allmaras and LES model), and a Dynamic Sub-Grid LES model.

The Spalart-Allmaras model is technically classified as a one-equation model (Spalart and Allmaras 1992). However, it internally solves for production, destruction, and dissipation effects and provides a resultant eddy viscosity corresponding to a Reynold's type stress. Additionally, it solves different model equations for different regions of the flow domain depending on the y+ variable (computed as a function of the flow solution).

Thus, Spalart-Allmaras gives fairly good overall behavior in both the near wall regions as well as the general, free shear flow domain. This has been validated with respect to the analytical solution of the free wall jet (Johnson and Bittorf 2002). It has been found to correctly predict results corresponding to the analytical solution for the entire flow domain (including vary large distances from the wall) for such quantities as the Jet Decay Exponent and similarity solution.

This is a major testament to the fidelity and accuracy of the solution of the original momentum and continuity equations. Another major characteristic is, that there are no tunable constants for this turbulence model other than the boundary conditions, which are zero for batch type mixing tanks, and have an inlet value for continuous flow processes that is generally set as a function of the Reynolds number.

The other characteristic is that since the model operates in both the near wall, and free flow domains, it can accommodate a more coarse boundary resolution, than models such as k-epsilon, which requires the first boundary layer elements to be in the y+ range of less than 5 or 10. Spalart-Allmaras can obtain reasonable results with the first element y+ on the order of 100 or 200. A direct benefit of this becomes apparent in the reduction of the computational model size and CPU time. Hence, Spalart-Allmaras is not as sensitive to "boundary layer resolution" issues as other models are.

The LES models and variants are for a different aspect of the flow analysis. LES by itself engenders a number of assumptions, which requires both a fairly fine resolution in both space and time everywhere in the model as a function of the desired frequency resolution. Generally, the LES is good in the "free stream" but not that good in the boundary layer. Hence, well-known research has developed a Detached Eddy Simulation (DES) model (Hughes et al 2000), which is a hybrid model that automatically applies a Spalart-Allmaras model for boundary layer regions, and the LES for the freer stream flow regions. LES is applied when the time varying characteristics of the pressure and flow are critical.

CFD Considerations and Modeling Approach

The approach used was to construct good approximate geometric models of generally available mixing impellers and internals, preserving the key functional characteristics, in the context of the selected 10 inch diameter Schedule 40 (254mm) piping and applicable fluid flow ranges. This is in contrast to having exact vendor specific drawings and dimensions and process specific flow rates.

Several modeling considerations were driven by the characteristics of the anticipated flow and to remove as many assumptions and physics approximations as possible. For example, in the case with the baffled orifice plate the entry flow to the mixer impinges on the equivalent of a forward facing step, and the exit region is basically a flow over a backward facing step. From a boundary condition standpoint, many assumptions and approximations would be required to specify the entry conditions to the mixer as well as loosing physics information regarding the flow pressure losses on the flat plate impingement if the entry region is not modeled correctly. Similarly, if the exit region is modeled as only the exit of the mixer, yet again, important flow physics is lost as well as fundamental developing mixing characteristics of swirling primary flow and the flow re-attachment to the pipe wall.

Further, from a pure mathematical / physics consideration, there is a major reason that AcuSolve correctly handles the appropriate flow physics, where other major codes have difficulty. That is, that in both the entry and exit regions, the flow has geometric volumes that are either convection or diffusion dominated. AcuSolve has the ability to handle both domains and their interface regions continuously. These are both hyperbolic and parabolic mathematical domains, which are very difficult to handle simultaneously and remain stable. This is uniquely handled through the internal Tau parameter of the Galerkin/Least-Squares formulation, which has the proven mathematical stability and convergence previously referenced. Most commercial programs can handle either the diffusion dominated or the convection dominated flows, but not both simultaneously. Hence, there are separate programs that are known to do well in Stokes (slow) flows or higher velocity Navier-Stokes or perhaps Euler flows.

The question for this becomes twofold. First, when is one program or the other, appropriate in flows that are on the boundary between the two flow domains and which of the two classes of programs should be selected when neither does very well for that gray area?

Second, many and perhaps most flows, such as the one encountered here, actually have both classes of flow within the model. Hence, neither of the above separate programs can deal effectively with the other effects so the solutions will be much less than optimal regarding the physics. This effect also becomes purposefully obscured with turbulence models, which attempt to introduce diffusion to make up for less than optimal implementations of the physics.

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