Advanced Fluid Dynamics
School of Cozmputing, Engineering and Mathematics
2014 ME335 CFD report
Flow in a turnaround duct
Eisa A M A Bandar
In this report, we will be able to learn and understand computational fluid dynamics, CFD and its description on numerical codes that help in determining the fluid properties like temperature, pressure, chemical composition and velocity. Additionally, this report provides answers relating to understanding phoenics library exploration; studying polis; flow in a duct in a turnaround and running tutorials. In the event of studying this report, students will be in position to understand phoenics as well understanding how effective and efficient the system can operate. In the phoenics process exploration, one will be in a position to explore phoenics, where in this case, pressure distribution, pressure distribution and view angles are of importance. From the tutorials, we are able to work and analyse on the retribution results. In fluid process flow, it is critical to understand that, fluid flow is dependent on the condition of the duct and is important for modelling and analysis.
As the term PHOENICS stands, it means Parabolic Hyperbolic of Elliptic Numerical Integration Code Series. It is therefore a general software used in analysis and understanding on how fluids like water, air, steam and flow in engines. In addition, this system is important when analysing physical and chemical composition. Quite a number of professional engineers and scientists apply PHOENICS when designing and making project plans. These projects may involve architectural designs for buildings, and engineering power designs. On the other hand, environmental specialists use phoenics in studying and controlling the environmental impacts as well as hazards. Deliver their jobs for example interpretation of experimental observations by scientists.
To answer the study questions on turbulence and boundary conditions, there were three consecutive mathematical models used and they were based on the studies.
1.1 Mathematical basis
The basic balance or conservation equation is:
Outflow from cell – Inflow into cell = Net source within cell
The quantities being balanced are the dependent variables: mass of a phase, mass of a chemical species, energy. The terms appearing in the balance equation are: Convection (that is directed mass flow), Diffusion (that is random motion of electrons, molecules or larger structures e.g. eddy).
To close the equation set, auxiliary equations are provided for the purposes of:
There may also be ‘artificial’ auxiliary equations, such as false transients (for relaxation), and boundary conditions. Examples of balance and auxiliary equations are
Dependent variables are the subject of a conservation equation yet auxiliary are constant, or derived from an algebraic expression.
These variables can be further subdivided into scalar and vector quantities in each case for example as shown below:
Scalars: pressure; temperature; enthalpy; mass fractions; volume fractions; turbulence quantities and various potentials.
Vectors: velocity resolutes; radiation fluxes and displacements.
Scalars: density; viscosity; conductivity; diffusivity; specific heat; thermal expansion coefficient; inter-fluid transport; absorptivity and compressibility.
Vectors: various non-isotropic properties; gravity forces and other body forces
The quantities defining the problem geometry can also be divided into scalar and vector categories:
Scalars: Inter-fluid surface area per unit volume; cell volumes and volume porosity factors
Vectors: Cell area porosities; cell centre coordinates; cell corner coordinates; centre to centre distances and cell surface areas.
Scalars are stored at the centre points of six-sided cells, with values supposed to be typical of the whole cell and Vectors are stored at the centre points of the six cell faces.
The following are the types of grid used in Phoenics: Cylindrical-polar; body fitted, orthogonal or non-orthogonal and cartesian. In all these cases, the grid distribution can be non-uniform in all coordinate directions. For cylindrical-polar coordinates, the following orientation is used:
The basic form of the balance equation is:
Outflow from cell – Inflow into cell = net source within cell
The quantities being balanced are the dependent variables from the earlier panel and examples of them are: energy; mass of a phase; mass of a chemical species; turbulence quantities; electric charge and momentum.
The single phase conservation equation solved by PHOENICS’s generalised form is:
This is an example of a particular derived equation:
With respect of the above equation are the turbulent and laminar viscosities, and Prt, Prl are the turbulent and laminar Prandtl/Schmidt Numbers.
After integration, the FVE has the form:
Where: (By continuity)
The neighbour links, the a’s, have the form
Convection diffusion transient.
The coefficients of this equation are obtained by plagging into correction form before the solution. In the correction form the sources are replaced by the errors in the real equation and the coefficients are only approximates. When the corrections tend to zero it means convergence is approached thereby reducing the possibility of round-off errors affecting the solution.
The neighbour links. This increases with the cell area, fluid density, transport coefficient and inflow velocity. On the other hand it decreases with intermodal distance. And in both instances are always positive.
The auxiliary equations are used to close the equation set and to provide for:
There may also be ‘artificial’ auxiliary equations, such as
The differential equations used need to be supplemented by boundary conditions before they are solved and the following are boundary conditions which define a flow problem: fixed flux; fixed value; linear and non-linear.
In PHOENICS the boundary conditions are represented as linearized sources for cells adjacent to boundaries as follows:
aBC is termed the COEFFICIENT.
is termed the VALUE.
aBC is added to aP, and is added to the RHS of the equation for
For fixed value boundary conditions
Considering a practical issue for example fixing temperature in one corner of a cube to 0.0, and 1.0 in diagonally opposite corner. The value of phi is fixed for any cell by setting C to a large number and V is kept to the required value. Taking that into consideration the equation becomes:
For fixed flux boundary condition
Considering when heat is generated at fixed rate. The fixed source is put into the equation by setting C a small number so that the denominator is not changed and setting V to (source/C). With this regard T ensures that the final source is per cell and the equation becomes:
By default all domain edges are impervious to flow, frictionless and adiabatic and they represent symmetry planes or axes.
The differential equation is integrated over a control volume to give the finite-volume equation that can be actually solved. The integral of the boundary source is represented in linearized algebraic expression form as below:
The finite-volume discretization of the differential equation thus yields, for each cell P in the domain, the following algebraic equation:
In respect of the equation:
S is the ‘true’ source
C is the coefficient
V is the value and T is the type, a geometrical multiplier
The following are boundary condition settings which are used in the CFD codes: fixed flux; fixed source; fixed value; linear boundary condition; non-linear boundary condition; inflows and outflows; wall conditions and general sources.
A turbulent flow is a fluid motion which is unsteady and irregular in space and time. It is always rotational and has high values of Reynolds number. Is three dimensional even when the mean is two dimensional? Turbulent flow also characterised by converting energy into heat because of viscous stresses
It is a calculation where it is possible in principle to simulate any turbulent flow by solving the foregoing exact equations with appropriate boundary conditions using suitable numerical procedures such as embodied in Phoenics. Examples of these equations are as follows:
For an incompressible flow
The equations for a scalar quantity C, such as species concentration, and an energy variable, such as enthalpy h, are:
LES stands for LARGE-EDDY SIMULATION, and is one of the two routes of predicting turbulent flow. The advantages of LES come from the fact that as large eddies are hard to model in a universal way they are simulated directly.
The application of LES to practical flows has been limited because of: prohibitive expense at high Reynolds number; difficulties in specifying initial and boundary conditions; and the need to perform 3D time-dependent simulations, even if the flow is 2D and statistically stationary.
Engineers are not concerned with all the details of the turbulent motion, but rather with its effects on the gross properties of the flow. Consequently, there is no need to solve for the INSTANTANEOUS variables if AVERAGED variables are all that is required.
The instantaneous variables are decomposed into MEAN and FLUCTUATING quantities:
Where the mean values are obtained by averaging over a time scale, dt, which is long compared to that of turbulent motion and in unsteady problems smaller compared with the unsteadiness state of the mean motion.
It is widely accepted that the Navier-Stokes (NS) equations together with the continuity equation comprise a closed set of equations and the solution of which provides a valid description of laminar and turbulent flows. Statistical-averaging process introduced an unknown turbulent correlation into the mean-flow equations which represent turbulent transport of momentum, heat and mass.
The difference between the eddy-viscosity and Reynolds stress model is that:
In EDDY-VISCOSITY MODELS, the unknown correlations are assumed to be proportional to the spatial gradients of the quantity they are meant to transport.
In REYNOLDS-STRESS MODELS, the unknown correlations are determined directly from the solution of differential transport equations in which they are the dependent variables.
This is because Vs and Ls are calculated directly from the local mean flow quantities where Vs is typical velocity scale and Ls typical length scale. The model is very simple and convenient to use whilst the main features of the simulation are being put together. A more accurate turbulence model can be activated once you are satisfied that other aspects of the flow are well represented.
Question Body-fitted coordinates:
BFCs are particularly suitable for internal or external flows with smoothly-varying non-regular boundaries. BFC can also be used to reduce numerical “false-diffusion” errors, by aligning the grid with the local flow direction where possible.
The main advantages of BFC are: toprovide good geometric representation; provides possibility of economical grid refinement close to the surface, and provides good representation of surface boundary layers, and hence of wall friction and heat transfer. . BFC also reduces numerical “false-diffusion” errors, by aligning the grid with the local flow direction where possible.
The convention for cell counting in BFC is the same as in Cartesian and polar geometries:
The locations of the grid corner points must be prescribed by the user, via a Cartesian frame of axes. The locations of the grid corner points are prescribed via a Cartesian frame of axes (XC, YC, and ZC). The alignment of the Cartesian axes and the origin can be chosen arbitrarily.
IX-direction this can also be called (I)
IY-direction this can also be called (J)
IZ-direction this can also be called (K)
Therefore there is no difference between (IX, IY, IZ) and (I, J, K) and are used to count cells in diverse directions. (XC, YC, CZ) are the Cartesian coordinates and specify the locations of the grid points
The relationship between the number of cells and the number of corner points in
Phoenics is that due to the finite-volume nature of PHOENICS, there are:
So there are more corner points than cells.
A BFC grid is defined via the CORNER POINTS where the cell faces intersect. The corner points are specified by their CARTESIAN COORDINATES.
They are stored at the centers of cell faces.
The fundamentals of BFC grid generation are:
Define Nodes – The Cartesian Coordinates of all grid nodes (cell corners) must be specified. Grid Orthogonally- Lines joining cell centers should intersect the dividing face at angles as close to right angles as possible. Angles less than about 30 degrees should be avoided if possible. This can be checked by turning the Grid check (ON) on the Match Grid dialog. Grid Spacing – Grid nodes should be spaced finely where flow properties are expected to change rapidly with distance – coarsely where they do not and smooth changes of aspect ratio are advised.
The process of specifying a BFC grid involves:
The inlet fluid goes into the system with consideration of the system inlet conditions. This flow is through domains towards the direction of the flow vector. During this continued flow process and as the fluid passes the domain wedge, it is guided to follow the colour, where the red section indicates velocity increase. On the other hand, the as the cross sectional domain decreases, the domain shift to the section with lowest cross-sectional area where it attains maximum velocity. The wedge size decreases as the flow continues, this way the cross-sectional area increases, decreasing the velocity along flow pipes until flow velocity in equalized in the entire system. Using the Bernoulli principle, it is clear that fluid flow velocity increases this in turn it was started by Bernoulli. Therefore, it should be understood that, constriction of the flow path reduced the Ericson system on energy density.
In this paper, we are developing a clear understanding to the readers to develop a analytical basis of the array creating duplicates as well as studying the surrounding under which the Phoenics operate. Moreover, this guide will help in duplicating one object and producing numerous arrays that relate to the same object. One cylinder was developed using the VR while the other five cylinders were made replicas using the array function. In this system, a velocity of 0.1ms-1, which was based on turbulence model.
It is appropriate to use cylinders since it gives uniform stream. This is because it has been divided into opposing directions, where one flow under one cylinder and the other flows over the second cylinder. The two flows are connected such that they pass through downstream of the cylinder and their flow is symetric and made with respect to the X-axis.
The section with high pressure is made infront od the cylinder, which is careated by stagnation point towards the surface that is near to it due to collinding flows into the cylinder.
A second stagnation point is created at the section where the flow from the cylinder, which is on the trailing surface. The two fluids flowing in the same direction below and above the stagnation point around cylinder meet at the trailing stagnation point. However, as the fluids move forward an enlargement space is created making the velocity constant thus making it normal flow.
On the other hand, as the cylinder outlet narrows, velocity of the flowing fluids increases. Similarly, as the space of the exit is reduced creates a restriction to the flow thus creating flow acceleration. As the separation of the two fluid increase between the bottom and the top corners, this reduces the velocity at those points. This separation of the flow it develops a reverse flow that produces flow eddies that are in the reverse. Due to these differences, there is significantly low-pressure difference.
It is easy to determine if the flow is turbulent or laminar by the use of Reynolds number ( ). Using the Reynolds number, if , the flow is said to laminar and if Reynolds number, then, the flow is turbulent and if the flow is more than 4000 it is turbulent flow.
The following equation was used to get the Reynolds number:
The table below is used to help in investigating inlet velocities range from 0.001m/s to 100 m/s and it compares the turbulent and laminar flows of the fluid using Reynolds number.