STORM Solver Overview

STORM is a general purpose computer program designed to numerically solve the Navier-Stokes equations, which consist of conservation equations for mass, momentum, and energy.  In addition, it is capable of solving an arbitrary number of general transport equations.  STORM uses a finite-volume representation of the governing equations, whereby the continuous problem domain is decomposed into multiple control volumes, and the governing equations are applied to individual control volumes and integrated over the entire computational domain.  This algebraic equation set is then solved using general and efficient numerical methods to obtain a solution of the engineering system. 

To obtain the greatest possible accuracy, speed, efficiency, and flexibility from STORM, Adaptive Research designed and integrated a number of innovative techniques into the program.

The PISO (Pressure Implicit with Splitting of Operators) algorithm used by the STORM Solver produces time-accurate calculation results when simulating transient phenomena; results superior to those produced by older finite volume algorithms. To achieve steady-state condition, PISO simply neglects the transient behavior, and marches in large time steps towards the converged solution.  A unique algorithm splitting process, coupled with an implicit scheme, makes STORM more computationally efficient, less memory intensive, and more flexible than other finite volume methodologies.

With STORM Solver, you may choose between Cartesian, Cylindrical-polar, or Body-Fitted coordinate systems to get the power and flexibility demanded by a particular problem. A mesh generated from body-fitted coordinates more exactly represents geometry contours and allows more accurate treatment of boundary and surface conditions in the model. Cartesian and cylindrical systems more simply represent geometry; and can save time in simulations involving less complex geometries or qualitative flow analyses.

STORM Solver lets you select from first, second, or third-order convection schemes, or a hybrid method. For problems where convection is negligible or the transport equations do not have convection terms, the solver provides the option to turn the terms off. The second and third-order schemes produce results with higher accuracy.

Finite-volume treatment of equations in general curvilinear coordinates produces accurate results on any smoothly varying grid-even in the presence of significant non-orthogonality. Using the integral form of equations, the finite-volume methodology enforces conservation.

In some applications, mesh sequencing improves convergence rates by an order of magnitude or more. STORM Solver provides an interactive mesh density control.  With it, you can coarsen or refine the grid during program execution in any or all coordinate directions In response, STORM refines the grid, updates boundary conditions, monitors point location, and interpolates the field solution to the new grid-automatically.

Solution Monitoring

The STORM Solution Monitor program provides “instant feedback” from the STORM solver regarding the status of your solution during execution. With this data, you can “fine tune” the solver parameters and therefore achieve the most efficient solution convergence. Both solution residuals and computed values of the dependent variables (“spot values”) are displayed on the Solution Monitor. Key solver parameters can also be "dynamically" adjusted, thereby avoiding the trouble and expense of halting your solution, returning to the CFD2000 interface, making the required changes, and then restarting your simulation.

Residuals provide a qualitative measure of convergence, and are computed by summing the current errors for each equation over the entire computational domain. As such, they provide an “at a glance” picture of the global state of the solution.

For a converging solution, the residual traces for all variables should generally decrease in magnitude as a function of time, with a rate that often becomes linear when plotted on the default logarithmic scale. Once the solution has fully converged, the residual plots traces should become fairly “flat,” with a very small incremental change (as indicated by the values listed under the Change column to the right of the plot) between updates.

The Runtime Spot-Value Plot shows color-coded traces of the instantaneous values of each active dependent variable from the start of the current run to the most recent update (usually, the current time step). Like the Runtime Residual Plot described above, the ordinate of this plot represents time, and the abscissa shows the computed values of the individual variables normalized by the quantities listed under Value.

Spot value monitoring is particularly useful for observing portions of your flow domain that may be responsible for hindering solution convergence. Examples of such potential problem areas include regions just upstream of any flow outlets and the “turnaround” or vortex shedding regions located downstream of flow obstacles. You can also use this technique as a “virtual measurement instrument” to observe the dynamic evolution of a particular variable of interest.

Turbulence

Turbulence modeling in STORM is accomplished using a two-equation k-epsilon model. This model solves transport equations for the turbulence kinetic energy k, and the dissipation rate e. The turbulent shear stresses in the Reynolds-averaged Navier-Stokes equations are then modeled using the Boussinesq hypothesis with an appropriate relation for the eddy or turbulent viscosity, based on the computed values of k and e.

STORM/CFD2000 provides seven (7) turbulence models:

           Standard k-epsilon  model

           RNG k-epsilon  model

           Chen-Kim k-epsilon   model

           Standard k-epsilon  model for low Reynolds number flows

           RNG k-epsilon  model for low-Reynolds number flows

           Chen-Kim k-epsilon  model for low Reynolds number flows

           LES - Smagorinsky's model

The Standard k-epsilon Turbulence Model

The k-e turbulence model is one of several two-equation models that have developed over the years.  It is probably the most widely and thoroughly tested of them all (Nallasamy, 1987). Based on simple dimensional arguments concerning the relationship between the size and the energetics of individual eddies in fully -developed, isotropic turbulence, the model employs the following diagnostic equation for the turbulent viscosity (Launder and Spalding, 1974).

where Cm is a dimensionless model constant, r is the local fluid density, and k and e are the specific turbulent kinetic energy (SI units: m²/s²) and turbulent kinetic energy dissipation rate (SI units: m²/s³), respectively.  These quantities are in turn computed using a pair of auxiliary transport equations of the form

where C1 and C2 are additional dimensionless model constants; Prk and Pre are the turbulent Prandtl numbers for kinetic energy and dissipation, respectively; Sk,p and Se,p are source terms for the kinetic energy and turbulent dissipation; and the turbulent production rate is

Chemistry

Chemical reaction models can be applied to situations involving combustion, power generation, propulsion systems, and chemical vapor deposition, among others. When considering chemical reaction simulations, two factors should be considered:

                                          the speed of the reaction, and

                                          the characteristic time scale of the flow itself

Reacting flows are classified according to the relative magnitudes of these time scales, and the analysis dictates which type of chemical reaction model should be used. STORM offers six types of reaction models with varying degrees of complexity and generality.

Full finite-rate, multi-step chemistry model with kinetic source terms. The most general of all STORM reaction models; useful for situations where reaction and flow time scales are comparable.

Specialized finite-rate, single-step chemistry model designed primarily for oxidation/combustion simulations.

"Fast chemistry" mixture fraction model for situations where the reaction time scale is much less than that of the flow.

Specialized "fast chemistry" model for two-way reactions; products based solely on thermodynamic state of the system.

Solves for chemical species transport without reaction (mixing only).

Free Surface Flow

There are many problems in fluid dynamics that involve the analysis of two-phase flows.  These flows are characterized by fluids which have large disparities in density and their relative motion is the essence of the multi-phase phenomenon.  Fluid surface tension plays a dominant role in the manner in which the fluids interact and largely determines the nature of the interface between fluids. Examples include droplets dynamics, tank sloshing, capillary motion, hydrodynamic stability and many others.

The Free Surface option provided in STORM predicts the motion of fluid interfaces based on the solution of a conservative transport equation for the fractional volume of fluid (VOF) defined as

Where V represents the volume occupied by the fluid within the control volume under consideration.  The function F obeys the equation

The solution to which provides information on the position and shape of the interface.  The local mixture density is then computed by

The momentum equation is also modified to contain an additional source terms relating to gravity, the curvature of the interface, and the fluid surface tension [1].  When the equation for F is solved within a computational cell, changes in F within the cell are recast as fluxes of  F across the cell faces.  To preserve the sharp definition of the free surface, a high-order TVD scheme [2] with damping is used.

Lagrangian Multi-Phase Flows

Many scientific and engineering problems can benefit from the simulation of two-phase flows. Examples include the design of power generating devices (such as internal combustion engines and liquid or solid rocket engines), pollution control equipment, nozzle designs, filter designs, and more.  In each application, the two-way coupling of the transportation of momentum, heat, and mass between the continuous phase and particulate phase plays an important role in the behavior of these flows.

STORM's advanced capabilities and features permit rigorous modeling of both flow and particulate behavior in complex, interacting fluid-particle phenomena.

       Liquid particles in a gas continuous phase

       Solid particles in a gas continuous phase

       Liquid particles in a liquid continuous phase

       Solid particles in a liquid continuous phase

       Gas bubble in a liquid continuous phase

Simulating two-phase flows that involve particle tracking requires the Lagrangian methodology because of its accuracy in calculating fluid-particle interaction between a dispersed phase and a continuous fluid. To track the particles in a realistic manner, the Lagrangian methodology directly models the physics of particle behavior in conjunction with the flow field.  It treats the particles as discrete entities in the flow field and calculates their relative trajectories. It then simultaneously describes, and consequently solves, the continuous phase using an Eulerian approach.  The flow field may be laminar or turbulent.

The Lagrangian methodology couples the solution of the dispersed phase to the continuous phase by representing the dispersed phase as a finite number of computational particles.  Then, the mass, momentum, and heat exchanges are computed between the two phases.  The Eulerian phase takes the exchanged amounts and adds them to the source term of the governing equations to quantify the effects of the dispersed phase. The particulate phase takes the exchanged amounts and calculates the particle characteristics (velocities and positions). Interactions between particles and particles with walls are modeled as well.

STORM's Lagrangian method makes it easy for the investigator to describe how the particulate phase is injected into the computational domain.  STORM provides detailed injection parameters that describe various planar or conical particle injection patterns. Conical injection patterns may be solid-cone or hollow-cone geometries, and can be applied at any arbitrary angle. Another injection parameter addresses non-uniform particle size distributions, which are often encountered in real-world particulate flow problems.

STORM also includes sophisticated “switch-on” physical models that permit more accurate description of the complex interactions between the particulate phase and continuous phases. These models further describe specific, observable particle behaviors, such as evaporation, breakup, combustion, and turbulent dispersion.  CFD2000 also accounts for complex particle collision behavior based on particle-to-particle collision and/or particle-to-boundary collision, including the effects of sticking and bouncing to boundary surfaces.

Compressible Flow

Analysis of unsteady compressible flow problems invariably involves the resolution of shock waves internal to the computational domain. Generally, these kind of problems typically difficult due to interaction between shock waves, expansion waves, contact discontinuities, as well as well as interaction between waves and wall boundaries. The development of high resolution numerical schemes, TVD schemes (Total Variation Diminishing methodology), has been a significant milestone for approaching the unsteady compressible flow problem.

STORM provides for the calculation of transient flow phenomena where the fluid is considered to be compressible. The Total Variation Diminishing (TVD) methodology has been implemented. The classic shock tube problem is illustrated in the following graph, providing a comparison between the analytical solution and numerical results.

Heat Transfer

Heat transfer through fluids, solids, and fluid-solid interfaces is modeled by STORM.  When heat transfer is being solved, the energy conservation equation solver is activated.

In many flow problems, there are solid objects within the computational domain. Though fluid cannot penetrate the solid-fluid interface, heat can be transferred through the interface and conducted inside the solid objects. In this circumstance, mass and momentum equations are solved in the fluid side only, but the energy equation is applied to both the fluid and solid regions. Because the solid-fluid interface requires attention to ensure appropriate conservation of energy, conjugate heat transfer analysis was developed.

Conjugate Heat Transfer

At the solid-fluid interface the following two conditions need to be met:

where the subscripts f and s are for fluid and solid respectively, and subscript i indicates interface. The two conditions state that the heat flux and temperature across the fluid-solid interface are continuous.

View Factor Radiative Heating Model

The view factor radiative heating model is a general-purpose routine for computing the exchange of thermal radiant energy between opaque surfaces. The view factor model is applicable to any geometric configuration, regardless of the orientation and curvature of the radiating surfaces, and regardless of the presence of internal or external blockages.

The view factor radiation model is well suited for any application in which the thermal contrasts within your flow domain are large enough to warrant the inclusion of radiative heating as an important form of internal energy transfer. It can be used alone or in combination with the conjugate heat transfer model to investigate a number of classes of problems, including:

Evaluation of heating, air conditioning, and ventilation systems in residences, auditoriums, or other commercial or public buildings

Computation of the airflow and temperatures in furnaces, ovens, or similar heated chambers

Evaluation of insulation schemes for heated (or cooled) pipes, ducts, or structural components (e.g., walls)

Heat exchanger design for power production, waste heat recovery, and chemical processing applications

Heat sensor element design analyses

Mutual energy exchanges between N radiating surface elements in an arbitrary flow domain

The view factor radiative heating model is based on a consideration of the net radiant energy exchange between individual computational elements lying on the interior surfaces of model geometries. These elements—which correspond exactly to the bounding walls of individual computational cells—are usually rectangular in shape, but can be of any size and orientation relative to each other. They can also be composed of different materials, have different temperatures, and absorb and emit thermal radiation differently by virtue of having different radiative properties.

The model consists of several components: (1) a model for the surface radiative energy flux and radiosity, (2) the view factor formulation, (3) a radiative obstruction model, (4) a set of simplifying assumptions, and (5) the energy conservation equation source term.

Moving Body Treatment

There are many CFD problems that can not be simulated using a fixed computational mesh. Such problems typically involve flows around moving or deforming bodies.  Industrial examples of these flows include flows in pipes with moving valves, flows in combustion chambers with moving pistons, the classical store separation problem, and many others.

The moving mesh capabilities in STORM/CFD2000 allow for the simulation of problems in which a body can move via translation, rotation, or simultaneous rotation and translation.

When the computational mesh moves as a function of time, the mesh velocity enters the analysis and must be included in discretizing the governing differential equations. Basically, the mesh motion affects the convective fluxes of mass, momentum, energy, and other scalar dependent variables.  In integral form, the continuity and the generalized transport equations can be written as follows:

Continuity Equation

General Transport Equation

where V is an arbitrary moving volume, A is the surface of V,  is any scalar quantity,  and S are the diffusive flux and source terms for the corresponding variable.

After the characteristics of the grid motion are specified, the mass and other convective fluxes across the cell faces are calculated according to the local fluid flow conditions and grid velocity. The cell volume, face area, and face direction cosines are recalculated at every time step.

Property and Physical Models

STORM can simulate a wide variety of gases and liquids, as well as their thermal interactions with many solids.  These materials are distinguished from one another by several parameters that characterize their distinctive properties.  These include (for fluids) the density, laminar viscosity, the specific heat, the thermal conductivity (heat transfer option specified), and the thermal expansion coefficient (incompressible fluid option specified).  The methodologies available in STORM for modeling each of these quantities are discussed below.

There are a number of choices to model the density, which may either be specified as a constant value, or prescribed as a function of temperature and pressure, or determined through the use of a custom model.  The options available are:  Constant Density, Ideal Gas Law, Isentropic Gas Law, and customizable density models.

The laminar viscosity coefficient controls the rate at which momentum is redistributed within the fluid due to molecular (i.e., diffusive) motions.  It is an intrinsic fluid property whose value specifies the correlation between the applied tangential stress on the fluid and the resulting rate of shear (deformation).  Both Newtonian and non-Newtonian models are available.  Non-Newtonian models include:   Power Law, Carreau, Bingham, Casson, and customizable viscosity models.

The specific heat at constant pressure Cp is an intrinsic material property (SI units: J/kg/deg K) which must be specified whenever the heat transfer option is selected. The options available are:  Constant Value, Field Value (e.g., for multi-species flows), and customizable specific heat models.

The thermal conductivity k is an intrinsic material property (SI units: w/m/deg K). There are  various options to define the thermal conductivity of a fluid or solid.  In many practical cases, the thermal conductivity is either a constant or a function of temperature.

STORM assigns fixed default values for this parameter valid at 300 K for most fluids.  The user may specify a value for the thermal expansion coefficient or decide to use the value for a particular fluid type from the Fluid Material Property Library.

STORM Solver Customization

The boundary condition options in STORM/CFD2000 allow the specification of constant values for the various dependent variables.  For some systems, it may be necessary to define these values as a function of the geometry, time, flow field etc.  This may be accomplished by choosing the "User Defined" option from the Boundary Conditions Specifications window.

CFD2000 includes a large library of fluid and solid properties listing the constant values for the thermo-physical properties of the respective fluids and solids.  For some systems, it may be necessary to define these values as a function of the geometry or the flow field.  For example, if the thermal conductivity of a material is a piece-wise linear function of the temperature, it will be necessary to use a look-up table to specify the proper value for the thermal conductivity.  This may be accomplished by choosing the "User Defined" option for the appropriate fluid or solid property from the Fluid/Solid Properties Specifications window.

STORM writes the results of the flow field simulation in a variety of formats.  In some situations, it may be desirable to have additional outputs, such as files that contain the maximum and minimum values of some parameter in a domain at each time step. Such customization may be accomplished by accessing the files "usercal.f" or "uoutput.f" through the Secondary File facility from the CFD2000 interface.