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 http://www.acusim.com/papers/index.html).
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.
Implementation
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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