Computational Fluid Dynamics - Math Problem Example

Computational Dynamics Name Lecturer Course Date 1). Mathematical models. a). Explain the physical meaning of each term of the Momentum Conservation equations. Momentum Conservation equation is for inertial and is written as (Fluent Inc, 2011) In this case i p represents static pressure that is pressure that is not changing Presents gravitational force on the body under inertia Presents forces that are being applied on the body Presents stress tensor which is calculated as In this equation is molecular viscosity and I is the unit tensor In axial and radial momentum conservation shown below =     +     +     + F x =     +     +     -     + F r (Fluent Inc, 2011) Where v z is the swirl velocity. b). Explain the reasons why CFD codes are written in low speed solver and high speed solvers. They are written in high speed and low speed codes depending on the objects that is being solved. If the object is high speed, then, high speed solver is used. A student is simulating an object flying at a velocity of 300m/s in the air using FDS5. Can this student obtain acceptable results and why? The student will not get reliable results because the object is flying at high speed and he has decided to use low speed FDS. c). the background pressure used in FDS - the background pressure used in FDS is P0 which is the atmosphere pressure. It is not possible for different rooms to use different background pressure if they are in the same level and in the same area. It is possible for different rooms to use different background pressure if they are not in the same area or level because background pressure is determined by the atmosphere of a certain area. 2). Turbulence a) The reasons why turbulence models are required in CFD – turbulence models are ideal in CFD problems because turbulence can be used to solve complex problems. This is because CFD models consist of 4 components which include grid and geometry generation, setting up a physical model, solving the model and eventually processing the solution. The CFD models involves the division of the compartment in numerous control volumes and through application of the model it is possible to calculate the heat and fluid flow occurring between each cell through use of fundamental physics equations. The number of cells which is used has a considerable effect on the results obtained. In is always desirable to have a large number of cells but his would call for more computer resources which will translate to high expenses. Such a problem cannot be solved by ordinary models accept a model that can handle a problem in stages. Turbulence models are not identical in setting them up which makes them easier to use in CFD. Halving the size of the grid cells in each direction will result in the doubling of the run time for each dimension in space and time. Changing the dimensions a mesh from a resolution will men better results, however, the reduction in cell size will take sixteen times longer to run. The large Eddy Simulation (LES), - is simulation where CFD problem is computed in time varying turbulent flow and the model that is solved have sub grid scale motions. The aim of large Eddy Simulation is to minimise the dependency flow models. Reynolds-Averaged Navier- Stokes (RANS), - this are equations which solve turbulent fluid flow problem by averaging time and fluctuating quantities. The important when it comes to solving a problem that requires space and time averaging. It can be used to solve a problem that has stationery turbulence, it is homogenous and requires a complex solution.. Direct Numerical Simulation (DNS) - is where a problem is solved numerically that is a solution of a continuity and N-S equations. This means a turbulent problem is solved numerically by using known time, length and velocity of the problem to be solved. In this case, time length and velocity will be used to set-up scales that will solve the problem. Details of LES used in FDS- large Eddy Simulation that is used in FDS software computes large vortexes without modelling them while it models small scale eddies. LES uses the following numerical parameters to solve a problem. ReL ReT N DNS NLES Ntime CPU 12300 360 6.7x 106 6.1x 105 2913 2.5 30800 800 4.0x 107 3.0x 106 3525 3.15 61600 1450 1.5x 108 1.0x 107 4200 3.73 230000 4650 2.1x 109 1.0x 108 54285 4.87 (Sodja, 2007) b). Based on Kolmogorov’s turbulence theory, explain the mesh resolution required by Direct Numerical Simulation and large Eddy Simulation. Mesh resolutions required by Direct Numerical Simulation and large Eddy Simulation assist in computational and they are selected based on their size and spacing. LES requires large domains with mesh spacing. Mesh spacing for large Eddy Simulation satisfies 50- 100. Direct Numerical Simulation does not require large mesh solutions because of the nature of their solutions. Critically compare their pros and cons The aim of Direct Numerical Simulation is find a solution for a turbulent velocity field without modelling. In other words a momentum problem will solved through calculations without modelling software, this will be very tedious. While Turbulence modelling estimates changed fluid stresses due to the swirling motion of a turbulent flow field. Direct Numerical Simulation code is very time consuming and has extensive storage requirements. Direct Numerical Simulation is important because it’s used to sturdy other models and it cannot be used to study turbulence or combustion interruption. The advantage of is that it is to reduce modelling impact and considers unsteady elements. Both Direct Numerical Simulation and large Eddy Simulation have disadvantages. One of the disadvantages of large Eddy Simulation is that it uses 3D simulations and sub-grid models. Direct Numerical Simulation requires numerical calculations and the use known data making it academic friendly. The zone model approach has been found to be the most commonly used as it is less demanding when computing and also is available at a lower cost. On the other hand FDS simulator that involve Large Eddy Simulations gives primary technique behind LES in which involves large tabulation of data that carries the bulk of the energy system and require direct resolutions so as to represent a flow process that has the desired accuracy. The modelling involving small scale eddies has in addition the ability to reduce computational demand with the overall speed of simulation process being improved. LES modelling do not require the use of the averaged parameters and this makes it possible for a transient solution being obtained easily 3). Combustion a). the premixed flame – is where fuel and oxygen are mixed and the flame starts after ignition. This type of flame has a uniform burning velocity due to the fact that it is determined by pressure, mixture ratio and temperature. This flame is as result of a burning gas from a specific container where the pressure of the gas in that container affects the velocity. This type of flame has high temperature because of the confinement of the burning substance. These types of flame will have an increasing flow rate when the nozzle is attached to the diffusion of the flame area, when the nozzle is far from the flame its rate increases. If this flame is lifted up then the structure of the flame becomes tri-brachial that is the flame is premixed in colour Premixed flames have very high temperature because they are limited in terms of air supply. The limited supply of air leads to the high heat release from flames. This type of flame is usually enclosed limiting the circulation of air thus ensuring the heat produced is high. The flame has a longer length because the burning substance which is mostly gas has a small opening which releases the burning substance. At times the flame length can fluctuate from time to time depending on the speed of air around the burning object. The speed of burning determines the length of the flame, if the speed is low then the length of burning flame. The trend in flame length behaviour is also determined by the fuel as well as the opening of the burning space. Gravity conditions determines also the length of flame during the burning period, the low gravity conditions leads to a smaller length unlike high gravity conditions. However the flame length is not consistent as it fluctuates from time to time. The flame length will up raptly reduce due to exhaustion of the burning substance. Where the gravity conditions are low but the opening is well structured, there is regular fluctuation in flame length. Diffusion flame –diffusion flame is when flame starts after oxygen and fuel mixes without ignition. In this case fuel is highly flammable and immediately it mixes with oxygen, it ignites. It produces turbulent fire which can be destructive to the environment. This type of flame cannot be easily have measurable parameter such as burning velocity since it is determined be the mixture of the burning substance and the oxidant. Such a type of flame is usually. The velocity of diffusion flame is high because the flame is trying to come out of enclosed environment in order to get air for burning. Diffusion flames have low velocity outside the burning area because of the air differences. The velocity of diffusion flames is determined using the momentum principle where mass flow rate plays an important role. When the mass rate is reducing, the flame velocity also reduces. The shape of diffusion flames is well defined because the burning fuel is ejected from a specific opening which has a certain gravity conditions. The gravity conditions determine the shape of the flame. Diffusion flame does not form huge soot as other flames because soot formation depends on the availability of air while the diffusion flame have limited supply of air. This ensures the volume soot that is produced during a burning is low. When the flow rate is high, then there is more soot that is formed. The major assumptions in the mixture fraction approach to simulate the diffusion flame – one of the assumptions is that there is equal diffusiveness in the chemical reactions. This assumption means that an equation of the mixture can be formed from equations of the mixture producing a conservative quantity. There is also an assumption of equilibrium chemical reactions. The major deficiencies caused by the assumptions – the assumptions have many deficiencies when it comes to lamina flows since turbulence flow does not work on molecular diffusions. Diffusion means that there is displacement from their original position through collision changing direction. b). mixture fraction Z is defined as Z= Where YF -mass fraction of fuel YO -mass fraction of oxygen -fuel mass fraction in the fuel stream - Mass fraction of oxygen at infinity S -ratio of oxygen and fuel molecular weight in stoichiometric mixture. It is known that = 0.95, = 0.23 and S = 0.64. Calculate Z on the flame (5marks). = answer Z= = Z= = 0.5642 If at the same position YF = 0.75, calculate z at this position(5marks). answer Z= = Z= = = 0.5489 4). Numerical Techniques a). procedures to numerically calculate ID unsteady heat conduction equation using explicit scheme – first you begin by reducing heat equations into initial and boundary conditions by identifying the flow of heart from a hot section to the cold section of the conducting element. The rate of heat conduction depends on the temperature difference Thot - Tcold , the length d, the cross-sectional area A through which heat flows, and the nature of the material itself. The greater the temperature difference, the greater the heat flow. The thicker the material, the longer it takes for the heat to travel through- since the energy transfer has to be passed along a longer chain of atomic collisions- making the rate of heat flow smaller. This is determined by the following equations. = 0 (1) The thermal conductivity of a substance is directly proportional to the rate at which energy is transferred through the substance. Higher values of k are associated with good conductors of heat, smaller values with thermal insulators that tend to prevent the flow of heat The dependence of the rate of heat flow through substance on all the factors mentioned is given by Fourier’s law of heat conduction: Where φ = Q/ is the rate of heat flow, k is the thermal conductivity of the material, A is the cross-sectional area, d is the thickness of the material, and T is the temperature difference between one side and the other. The quantity is called the temperature gradient; it tells how many OC or K the temperature changes per unit of distance moved along the path of heat flow. For any case ≈ (2) For explicit scheme = (3) Substituting 2 and into 1 to yield - k =0 =0 Let = , then the above equation can be changed to =0 (4) At time t=0 (i After one time step at t=, equation 4 can be written as (5) The equations will give values which is the temperature at each node at t=. The next step is at at t= thus having equation (4) as (6) This process is repeated for other figure to get temperatures to the lat node chosen For implicit scheme = 0 (1) For any case ≈ (2) For explicit scheme = (3) Substituting 2 and into 1 to yield - k =0 =0 Let = , then the above equation can be changed to =0 =0 =0 (4) At time t=0 (i After one time step at t=, equation 4 can be written as The equations will give values which are the temperature at each node at t=. b).consider a thin insulated rod 0.1 m long with k = 0.835 x 10-4 m2/s. let t = 0.1 seconds. At t=0 the temperature of the rod is zero. If one of the two ends is maintained at a temperature of zero and the other end is raised to a constant temperature 22oC immediately after 0s. calculate the temperature distribution at t = 1 sec (10 marks). The calculation of λ is a follows = , = = 0.021 and is less than 0.5 requiring the use of explicit method (5) The equations will give values which is the temperature at each node at t=. 18, 15, 13, 15 References Fluent Inc, 2011. Continuity and Momentum Equations. Available online at [ Accessed 22 April 2013]. Rudman,M. & Blackburn, H 1999. Large eddy simulation of turbulent pipe flow. Second International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia. Available online at [ Accessed 22 April 2013]. Sodja, J., 2007. Turbulence models in CFD. University of Ljubljana Faculty for mathematics and physics Department of physics. Available online at [ Accessed 22 April 2013]. Read More

Reynolds-Averaged Navier- Stokes (RANS), - this are equations which solve turbulent fluid flow problem by averaging time and fluctuating quantities. The important when it comes to solving a problem that requires space and time averaging. It can be used to solve a problem that has stationery turbulence, it is homogenous and requires a complex solution.. Direct Numerical Simulation (DNS) - is where a problem is solved numerically that is a solution of a continuity and N-S equations. This means a turbulent problem is solved numerically by using known time, length and velocity of the problem to be solved.

In this case, time length and velocity will be used to set-up scales that will solve the problem. Details of LES used in FDS- large Eddy Simulation that is used in FDS software computes large vortexes without modelling them while it models small scale eddies. LES uses the following numerical parameters to solve a problem. ReL ReT N DNS NLES Ntime CPU 12300 360 6.7x 106 6.1x 105 2913 2.5 30800 800 4.0x 107 3.0x 106 3525 3.15 61600 1450 1.5x 108 1.0x 107 4200 3.73 230000 4650 2.1x 109 1.0x 108 54285 4.

87 (Sodja, 2007) b). Based on Kolmogorov’s turbulence theory, explain the mesh resolution required by Direct Numerical Simulation and large Eddy Simulation. Mesh resolutions required by Direct Numerical Simulation and large Eddy Simulation assist in computational and they are selected based on their size and spacing. LES requires large domains with mesh spacing. Mesh spacing for large Eddy Simulation satisfies 50- 100. Direct Numerical Simulation does not require large mesh solutions because of the nature of their solutions.

Critically compare their pros and cons The aim of Direct Numerical Simulation is find a solution for a turbulent velocity field without modelling. In other words a momentum problem will solved through calculations without modelling software, this will be very tedious. While Turbulence modelling estimates changed fluid stresses due to the swirling motion of a turbulent flow field. Direct Numerical Simulation code is very time consuming and has extensive storage requirements. Direct Numerical Simulation is important because it’s used to sturdy other models and it cannot be used to study turbulence or combustion interruption.

The advantage of is that it is to reduce modelling impact and considers unsteady elements. Both Direct Numerical Simulation and large Eddy Simulation have disadvantages. One of the disadvantages of large Eddy Simulation is that it uses 3D simulations and sub-grid models. Direct Numerical Simulation requires numerical calculations and the use known data making it academic friendly. The zone model approach has been found to be the most commonly used as it is less demanding when computing and also is available at a lower cost.

On the other hand FDS simulator that involve Large Eddy Simulations gives primary technique behind LES in which involves large tabulation of data that carries the bulk of the energy system and require direct resolutions so as to represent a flow process that has the desired accuracy. The modelling involving small scale eddies has in addition the ability to reduce computational demand with the overall speed of simulation process being improved. LES modelling do not require the use of the averaged parameters and this makes it possible for a transient solution being obtained easily 3).

Combustion a). the premixed flame – is where fuel and oxygen are mixed and the flame starts after ignition. This type of flame has a uniform burning velocity due to the fact that it is determined by pressure, mixture ratio and temperature. This flame is as result of a burning gas from a specific container where the pressure of the gas in that container affects the velocity. This type of flame has high temperature because of the confinement of the burning substance. These types of flame will have an increasing flow rate when the nozzle is attached to the diffusion of the flame area, when the nozzle is far from the flame its rate increases.

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