Computational Fluid Dynamics: The Basics with Applications - Assignment Example

COMPUTATIONAL FLUID DYNAMICS Name Course Professor Date Question 1 (a) In any CFD simulation, the graphic representation is determined by the governing equation for the physical phenomenon. When the viscous effects are ignored and only incompressible flow is considered in a fire simulation, the following governing equation is obtained: Vt + (V.∇)V + = 0…………………… (1) This equation factors in various parameters in the fire including its instantaneous velocities (Vt), absolute velocity (V), partial derivatives of its absolute velocity (∇) and pressure (∇P), and the density of the gas in the flame (p). With these parameters, it is possible to develop a comprehensive simulation of the fire under study. Another governing equation that can be used for analysis of fire is the one shown below. It is referred to as the level set equation and is used in tracking the interface points as the fire moves from one point to another. φt + W · ∇φ = 0 ……………………… (2) The parameters that are included in the equation include the location of the interface at any particular time (φt), the velocity given by the partial derivative of the point of interface (∇φ), and the actual velocity of the flame (W) along the propagation axis. In order to solve both equation 1 and 2, all the boundary conditions must be considered. This is because the associated boundary value problems enable one to obtain a solution that factor in all the prevailing conditions at the boundaries as they are separated by space (Anderson, J., 80, 1995). These include the amount of fuel at one boundary or the quantity of heat generated at another. There are multiple boundary conditions in a fire simulation- some of these are: 1. Radiation boundary condition 2. Heat flux boundary condition 3. Convection boundary condition (b) In CFD analyses, the codes are written in low speed solvers when the phenomenon being modeled comprises of incompressible flow. This approach is mainly focused on the pressures involved in the flow as it is a function of neither temperature nor density. Solving the problem is solved using a pressure correction equation derived from the conservation of momentum and continuity equations. On the other hand, the codes are written in high speed solvers if the flow is compressible. This is because the pressure herein is determined by both the temperature and the density. Any given problem can therefore be solved using the continuity equation and equation of state. The student will be able to obtain acceptable results if they take the right approach. This should begin by developing the appropriate boundary value problem. This should include the right governing equations and boundary conditions to obtain the effectively solve the problem. Based on the speed the object is flying at, they should code should be written on a high speed solver. In this way, the results obtained will be an acceptable accurate simulation. (c) The background pressure used in FDS is the perturbation pressure. It is possible for two rooms to have different perturbation pressures because they could be having different physical conditions. These include things such as the room temperature, draft, and humidity levels. They could also have different dimensions and designs, thereby varying the background pressures also. The following formula shows that change in pressure is dependent on the height of a room (h). If this height varies in the two rooms, then the perturbation pressures will also differ. ∆P = Consequently, different pressure zones can be defined using FDS input instructions. These include the following ones which define the mesh parameters for two different rooms: 1. &MESH IJK = 50, 70, 90, XB = 0,12, 0,14, 0,16 2. &MESH IJK = 60, 50, 40, XB = 0,18, 0,14, 0,10 Question 2 (a) Turbulence models are some of the most important parts of CFD analysis. They are important for a number of reasons. One of these is that they enable the analyst predict extent of the turbulence and how it varies in different conditions. This makes it possible for them to study the critical conditions in an easy and accurate manner. There are some turbulent conditions that would pose grave danger if done on a prototype or were to occur during the operation of a machine. These can be safely and closely analyzed through CFD models and the relevant techniques of overcoming them can be overcome. These analyses are also very cost effective. When doing an extensive project, it would be much costlier and time consuming to build a working prototype. However, its operations can be effectively modeled and turbulence patterns determined using CFD applications. Some turbulence patterns are very complex and obtaining them on a prototype would be impossible. Since it is necessary to obtain the relevant turbulence information before going forward with developing the application, CFD modeling provides a viable option. The complex situations can therefore be analyzed from different perspectives to provide all the data that is needed (Huebner, K., Thornton, E. & Byron, T., 96, 1995). Large Eddy Simulations (LES) are 3D simulations for time-dependent very turbulent flows with large eddies arising from their high Reynolds numbers. It enables the modeling of large eddies that would be hard to model using other techniques. The Reynolds-Averaged Navier- Stokes (RANS) are those simulations which are primarily based on the Navier-Stokes laws as one of the governing equations and a mean of the prevailing Reynolds numbers as one of the determinants of the boundary conditions. This allows for the rationalization and modeling of flows with conditions that are continuously and rapidly fluctuating. A Direct Numerical Simulation (DNS) are those simulations whereby the Navier-Stokes equations are solved directly without developing any turbulence model first. Since it involves the resolution of all spatial and temporal parameters, it is useful in the analysis of flows along and between boundaries. There are different LES models that can be developed on the FDS6 application, each one having determining parameters. One of these is the Germano dynamic model, which is a function of the following parameters: 1. The Reynolds number- The value of this quantity determines the nature of flow being modeled. It can be laminar, transitional or turbulent. 2. Stress tensor- It is comprised of nine components which define the stress conditions within the fluid in the three axes of space 3. Kinematic viscosity- it is a dimensionless unit that determines how viscous a fluid is and therefore how easy it can flow 4. Temperature- this is the surface temperature of the fluid and has a direct bearing on all the other parameters. (b) Kolmogorov postulated that in any turbulent flow, the eddies are of different sizes. Their kinetic energies, flow velocities and times scales therefore vary from one another. The prevailing sizes therefore have to be considered when modeling the mesh resolutions for their velocity and length scales in both DNS and LES to prevent aliasing and blurring (Carslaw, H. & Jaeger, J. 209, 1959). When developing simulations based on LES, the major factor that determines how the big the resolutions are is the mesh size. There are a number of pros associated with this approach. One is that it simplifies the analysis as the meshes can be easily set in proportion with the eddy sizes. Additionally, the sizes can be scaled down to a degree where smaller layers of the simulation being resolved and hence providing more sophisticated analyses. However, it has some cons too. These include the fact that certain points can be overlooked in the simulation and hence the resolution will not be as accurate as it should be. In addition to this, the technique requires one to have extensive working knowledge of how the chosen mesh pattern will produce the best resolution. This means that an analyst who is not very experienced may not be able to produce the best results. If the same resolutions are developed in DNS, the number of nodes is the primary determinant. One major advantage with this technique is that an analyst can set as many of nodes as they desire to obtain the most critical of conditions. This allows for them to obtain highly differentiated sets of accurate results. Another pro associated with it is that the nodes can be aligned in any way thereby providing resolutions that factor in multiple prevailing conditions. Despite these benefits, it also has some drawbacks (Patankar, S., 119, 1980). Top among these is because that the nodes used may not be placed well enough to form a complete loop. The parts that are left out may therefore lead to inconsistencies in the resolution. The other con that is associated with it is that at times the nodes may be inconsistently distributed over the surface being analyzed. This causes the development of pressure zones which can affect the accuracy of the final resolution. Question 3 One of the most important things that are needed in a fire simulation is the combustion model for several reasons. The models indicate how and where the flame originates and hence enables the CFD analyst to develop a resolution that factors in the location and nature of the source. They are also necessary as they show the fuel consumption- especially in controlled fires. For a fire simulation to be truly accurate, the amount of fuel it consumes needs to be considered so as to provide the necessary parameters that are essential for further analysis. It is also a strong indicator of the chemical composition of the fire being modeled. Furthermore, they are also important indicators of the flame intensities for the fire being simulated. This is necessary as it is only through its quantity can its effects and movement be determined from the simulation. Making informed assumptions is essential in many mathematical and physical models to ease the relevant analyses. Based on the mixture fraction approach, there are some assumptions that are necessary during the resolution of an FDS. One is that the flame is only dependent on three factors: oxygen, fuel, and nitrogen. It is assumed that by tracking the time scales of these three can give a clear picture of the nature of the flame and hence the development of an FDS. This assumption comes about because these three components mix in fixed fractions which can be monitored and used in the simulations (Cant, R. & Mastorakos, E., 178, 2007). Another common assumption is that all the chemical species in a flame can be replaced by one scalar. This quantity represents the amount of oxygen and fuel that can be found in any location of the flame at any given time. This simplifies the simulation further as it minimizes the number of items being tracked as the flame burns. The final assumption made is that the reaction rate is infinite. This assumption ignores the fact that either the fuel or oxygen will be depleted and thus allows for the flame to be analyzed as if it would burn for a limitless period of time. A pre-mixed combustion and explosion can be simulated in FSD6. This is because the application offers a wide array of tools that allows the analyst to set the specific parameters that govern the flames in these fires. Therefore, the kinds of fires produced by such flames can be accurately modeled. (b) Z = Yo = 1-Y͚ = (1- 0.23) = 0.77 YF = 1- Y͚ = (1-0.93) = 0.07 Hence, Z = [0.63*0.07 – (0.77 – 0.23)]/ [0.63*0.93 + 0.23) = -0.61 From the law of conservation of energy, the different parameters are supposed to sum to either zero or one. Since they are integral values, the latter is considered and hence all the unknowns can be derived from manipulation of the given values and this summation (Launder, B., Spalding, D., 269, 1974). If YF = 0.68, Y͚ = 1-0.68 = 0.32 Z = [0.63*0.68 – (0.77 – 0.23)]/ [0.63*0.32 + 0.23) = -0.23 Question 4 (a) The 1-D unsteady heat equation is one of the most common CFD problems. It can be solved numerically using the explicit scheme as follows: 1. The initial and boundary conditions associated with it are first determined. Since the equation goes with one initial condition, a boundary condition is required at each location or point in the boundary. 2. The first partial order derivative of the equation with respect to time is then obtained and set at one boundary condition 3. After this, the second order derivative is obtained at the second boundary condition and multiplied with a constant referred to as the diffusion coefficient. It is also possible to solve the problem based on the explicit scheme. This is done in the following manner: 1. The discretization process begins by establishing a set of mesh points that are used to replace the domain [0, L] x [0, T]. These points should be equidistant from one another 2. The mesh functions are then determined and their approximate values at pre-selected points are calculated. The obtained values are checked for whether they satisfy the initial PDE 3. Using a central difference in space and a forward difference in time, the derivatives are replaced using these finite difference approximations 4. Once step 3 is done, the equation is transformed into its algebraic form which can be easily solved by placing the required values. (b) L = 0.125 m K = 0.85*10-4 m2/s ∆x = 0.025 m ∆t = 0.1s Ut=0 = 0oC UL=0.125 = 20oC Temperature distribution after 1 second = Ut=1 Heat equation: - = 0 Where Q = Therefore to obtain the distribution, this equation can be partially integrated with respect to time and multiplying with ∆xe by assuming 4 nodal points. This leads to: ∆xe -dx + Qdx = 0 Since Q is the only unknown, it can be resolved from the equation. And thus: [0.025ex0.125]-[( 0.85*10-4 )x 0.025e] = -0.025eQ Hence, Q = 125 J Therefore, at 0.1s and, U = 125 *0.1 = 12.5 oC. References Anderson, J., 1995. Computational Fluid Dynamics: The Basics With Applications. McGraw-Hill Science Cant, R. & Mastorakos, E., 2007. An Introduction to Turbulent Reacting Flows. London: Imperial College Press Carslaw, H. & Jaeger, J. 1959. Conduction of Heat in Solids (2nd ed.). Oxford University Press Huebner, K., Thornton, E. & Byron, T., 1995. The Finite Element Method for Engineers (Third ed.). Wiley Interscience. Launder, B., Spalding, D., 1974. "The Numerical Computation of Turbulent Flows". Computer Methods in Applied Mechanics and Engineering  Patankar, S., 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere Series on Computational Methods in Mechanics and Thermal Science. Taylor & Francis. Read More