Section A
Introduction:
Computational Fluid Dynamics (CFD) can defined as a set of numerical methods applied to obtain approximate solutions of problems of fluid dynamics and heat transfer. Through the relevant science-based mathematical equations and data structures provided, CFD can predict the fluid flow, transfer of heat or mass, chemical reactions, and related phenomena. Table below is showing the development of CFD:
Period Event
1920s The first numerical weather prediction system in the world was developed by Lewis Fry Richardson.
1930s
–
1950s In 1933, the earliest numerical solution for flow past a cylinder was be created.
By using the mechanical desk calculator, Kawaguti obtains a solution for the flow around the cylinder.
In 1957, the first functional CFD computer simulation model was developed by a team from Los Alamos National Lab. Many of numerical methods which created by the team was still using in today. Such as:-
k-ε turbulence model.
And more….
1960s
–
1970s Through by the computing technology develop in the 1950s. The practical of CFD came to realize, especially in aerospace industries.
The Aircraft makers was began focus on development of CDF and the technology.
In 1967, Douglas Aircraft had relatively perfected in 3 dimensional CFD analysis method.
Boeing was incorporated the full potential equations into the technology in the 1970.
111980s
–
1990s In 1981, the Eular equation for transonic flows was incorporated into codes.
Because of advance of the technology and computing ability.
In 1990s, the automakers was began seeing the application in automotive design. Such as :
GM (US) and Ford (US) began using the CFD in 1995.
2000s
–
now Commercial CFD codes originated that are available today:-
Fluent (UK and US)
CFD++ (US)
Flow 3D (US)
SCRYU (Japan)
….
Application of Computational Fluid Dynamics (CFD):
Computational Fluid Dynamics gives an insight about flow patterns, patterns was complex, difficult, and expensive by scientific process module which resolve different fluid flow related problems such as temperature, flow velocity, density and chemical concentrations in the figure where the flow is present. The Fluent software once of the tool was applied in the CFD. The fluent software is useful in the mechanical engineering or industries. (Aircraft and Automotive manufacture)
For example, CFD can used to make a research and understanding the interaction of propellers or rotors with the body of airplane fuselage when the airplane flying in the sky with different situation setup. (Example: velocity of wind)
Figure 1: The Simulation showing the pressure applied on airplane and the direction or pattern of vector flow on wing.
Throughout the CFD simulation, we can obtained clearly and understand how the pressure field applied on any position on the body or part of the airplane. CFD simulation also can used to simulate to direction or pattern of fluid flow through the parts of object in vector. Based on the result showed (Like figure 1 showed), the designer can understand the flow of the fluid, and they can modify the design to improve the efficiency of the equipment used.
Figure 2: The temperature distribution in jet engine.
The temperature distribution also can be obtain in the CFD simulation (Like figure above). CDF simulation also can used to obtain the temperature field, heat flow or the temperature changing in the objects. The effectiveness of a simpler manifold design was showed and verify without the need for field testing.
Of course, CFD simulation also can applied in different range of industries not in Automotive or aircraft only. It uses to obtain flawless designation of product with the combination of fluid dynamics’ theory. Like:-
Building heating
Ventilation
Air conditioning (HVAC)
Chemical manufacturer
Oil and gas industry
Transportation
Product design and optimization
Meteorology
Weather forecasting
Energy or power generation
Methodology:
The methodology of the computational fluid dynamics is basically categorised into 2 methods which is discretisation method and turbulence models.
The methods used by discretisation methods are finite difference method, finite volume method and finite element method. Finite difference method was the first utilised by Euler in 1768 and it was the simplest to the program. The finite difference method was directly applied to the differential form of the governing equation and consider the following example. To find the value of the first derivative of the scalar function U(x) at a fixed point x0. By Taylor series to solve U(x0 + ∆x):
U(x_0+∆x)= U(x_0 )+∆x ∂U/∂x |_(x_0 )+〖∆x〗^2/2 (∂^2 y)/(∂x^2 )+ …
Approximated as:
∂U/∂x |_(x_0 )=(U(x_0+∆x)- U(x_0 ))/∆x+O(∆x)
The advantage by using this method is its simplicity. It is the simplest method to use to obtain the result. Besides that, it able to achieve high-order accuracy of the spatial discretisation because it is easy to obtain the high-order approximations. Furthermore, the method requires a structured grid, the range of application is clearly restricted to avoid further problems. The disadvantage of this methods is it cannot be applied in body-fitted which is curvilinear coordinates and the governing equations have to be the first transformed into a Cartesian coordinates system or in other words is from the physical to the computational space which given in figure at next page.
Figure 3: From computational space to physical space.
The other method used is finite volume method (FVM), it is apply directly utilises the conservation laws of integral form of the Navier-Stokes to a control volume defined to get a discrete Euler equation for the defined control volume. The continuity equation integral form is only valid for steady, incompressible flow had shown below:
∫_s▒V .n dS=0
S will represent the surface of the control volume while n is the outward normal at the surface. This equation shows that the net volume flow into the control volume is zero (Rajesh Bhaskaran, 2014). 2 basic approaches can be categorised as:
Cell-centred scheme which the flows quantities stored at the centroids of the grid cell
Cell-vertex scheme which the flows quantities stored at the grid points
Figure 4: Cell-Centred and Vertex-Centred.
The mainly advantage of FVM is that the spatial discretisation can be directly carried out in the physical space without causing any transformation problems between physical and computational coordinate system.
Besides that, the method that used is finite element method, it is general applied to the Euler and Navier-Stokes equation solution, start with a subdivision of the physical space into triangular (in 2-D) or into tetrahedral (in 3-D) elements (Jiri Blazek, 2005). The degree of freedom which is the number of independent parameters that define its configuration and it was determined by the total number of points multiplied with the number unknown.
Figure 5: The variation of the solution inside an element will be represent by the shape function.
The finite element method has adopt for use with the governing equation from the differential into an equivalent integral form by using variational principle and weighted residuals or call weak formulation. The finite element method’s advantage is that it involve more mathematics, natural boundary conditions which is for fluxes, master element formulation and any shaped geometry can be modelled with the same effort. It involve too much mathematical which causing less physical significance.
Turbulence modelling allow the user to find the most suitable turbulence model to stimulate the turbulent flow to optimize the design in real world. The turbulence model include Spalart-Allmaras, Menter’s Shear Stress Transport, k-epsilon, k-omega and Reynolds stress equation model. K-epsilon is the family of Reynolds-averaged Navier-Stokes models. It is use to tracking the changing of k and epsilon while k is turbulent kinetic energy and epsilon is the rate of energy dissipation, or change in turbulent kinetic energy. The characteristic of the k-epsilon is relatively stable and converges smoothly. The k-epsilon models include no-slip wall, adverse pressure gradient, flow with strong curvature, rotating flow and it having difficulty in solving the epsilon.
Working principle:
Governing equation for fluid flow common used in Computational Fluid Dynamics (CFD) is the Navier-Stokes equation and conversation of laws. In the conservation of laws, it consists of continuity equation, momentum equation and energy equation.
General equation of conservation law of rate of total change written as
Where is a fixed control volume.
The continuity equation describes the conservation of mass
The momentum equation describes the conservation of momentum
The energy equation,
where is the total energy per unit volume.
Compressible Navier-Stokes equation in integral form;
Since
Then,
For incompressible fluid flow, constant
Continuity equation:
Momentum equation:
Energy equation does not included, since it does not momentum and continuity equation. Hence, the incompressible Navier-Stokes equation can simply written as:
Note that kinematic viscosity,
In CFD, generation of grip is separated of structured and unstructured from the generated volume grid inside the flow domain.
Structured grip
The structured grip can be solved by two different approaches which are algebraic or using partial differentiation equations (PDE’s). PDE’s of two common types are elliptic and hyperbolic equation.
Elliptic equation for 2D
Hyperbolic equation for 2D
Unstructured grip
The unstructured grip methodologies for CFD application are delaunay or advancing-front.
To improve the grip regularity or smoothing, Laplacian operator was used to move the grid nodes. The formula given as
Where =number of nodes adjacent to i and .
Advantages and Disadvantages:
By using the CFD software, the time and cost of the development is reduced until relatively low (Patel, 2013). During a development of new product such as car or an airplane, the starting-cost, running-cost and time usage to create a model or prototype simulation are extremely high compared to CFD simulation by computer setup. Most of the physical models are built in scale for the big size object but in CFD the object is built in the scale of 1:1. (Kevin W., Robert G. 2008) In CFD, the flow regime is nearly identical but in physical models the flow regime of the fluid is under control by many complex way just in order to bring the Reynolds number match with the theoretical value in the physical experiments. By using those physical experiments and tests to obtain essential data is not cost-effectively compared to CFD simulation. Consequently, the performance of the product can be foresaw to determine is the assets worth to invest into the development.
On the other hand, the CFD simulation can reduce the difficulties and dangerous to the lowest (Patel, 2013). This is due to the reason that there is a lot of problems and dangerous will be faced when the experiment is conducting with model and prototype. By using CFD software, it is able to simulate the physical condition of an object theoretically before it is adapting or executing to the real product of the development. Consequently, the problems and dangerous can be minimized and identified to increase the performance effectively and efficiently by analyzing the solution for future development.
Apart from that, the data can be extracted easily by going through CFD software. This is due to the reason that the data of the object in the physical experiments only can obtained in limited number of location by using sensors or gauges. On the contrary, the CFD software allow the operators to examine any location or region of the objects and yields a comprehensive set of thermal and flow parameters for examination (PRETechnologies, 2014). As a result, the data not only are able to extract more time-effectively and cost effectively, but more accurately compared to physical model experiments.
In contrast, although the CFD can simulate the flow condition, but there is not always the exact scenario in real world, there are lots of numerical error is introduced in CFD software which included round-off error and truncation error. The round off error is due to the finite word size available in the computer which is the difference of the approximation of a number and the exact value due to rounding (EW Weisstein, 1999). The truncation error usually happened due to approximation in numerical model. It will happen and go to zero as the grid is refined. A mesh refinement should be done in order to minimize the truncation error (Margaret Rouse, 2012). Consequently, the accuracy of the result may be doubted which will not always obtain a successful result.
On the other hand, the CFD software is not easy to understand and user friendly. This is due to the reason that it involve a lot of high end mathematics, physics and computer science, the untrained user may not working well with the software (Patel, 2013).. Additionally, the error is easier to be obtained due to the simple models or the simplified boundary condition. Besides, the most complex simulation must use more powerful supercomputer to do the calculation but not the normal PC due to it may require more compute time as it more complicated to set up. Furthermore, the CFD simulations also are limited by availability model which the non-mathematical phenomenon model is unlikely to be modelled in the CFD software.
Section B: (Part 1)
Basic Set Up for this Stimulation:
Geometry
Draw the geometry of the desired design according to the dimension given and sketch.
At the line x=0 is the line of sudden contraction occurs.
Figure 1.1: Geometry
Mesh
Insert the inflation of 20 layers to the wall. As a result, the more information we can get along the wall because the wall is where the boundary layer occurs. By inserting inflation, the detail information and more accurate data can be obtained. Besides, trial and error was done to look for the most suitable mesh size.
Minimum Size = 0.09mm. Maximum Size =0.19mm.
Figure 1.2: Mesh with mesh size = 0.09mm
Minimum Size = 0.02mm. Maximum Size =0.05mm.
Figure 1.3: Mesh with mesh size = 0.02mm
The mesh with mesh size of 0.02mm is more concentrated and more cell were generated. Thus, more precise or accurate data can be obtained with mesh size of 0.02mm.
Setup
Model selection
Select the desired model to stimulate the turbulence fluid flow to optimize the design in the real world based on the desired conditions. In our case, k-epsilon was selected.
Figure 1.4: Model Selection
Materials
Change the default fluid into water-liquid from the fluid database and observed that the density and viscosity of water is 998.2 kg/m3 and 0.001003 kg/(s·m).
Figure 1.5: Materials Selection
Cell zone condition
Figure 1.6: Cell Zone Condition
Boundary condition
Velocity
Input the magnitude to the velocity inlet.
Figure 1.7: Way to adjust the velocity of inlet
Pressure
Input the value to the pressure outlet.
Figure 1.8: Way to adjust the pressure of outlet
Solution Initialize
Initialize the solution by selecting the Hybrid initialization methods
Figure 1.9: Solution Initialization
Calculation
Adjust the iterations to the 1500 to obtain a converged result.
Figure 1.10: Way to Run Calculation
Mesh size determination
The mesh was determined to be 0.02mm in size and 0.05mm max size. The reason of choosing this value is because the value of 0.09mm min size and 0.19mm max size fluctuate in a large amount value comparing to the next value. The graph below shows the graph of pressure difference against mesh size.
Graph 1.1: Pressure difference against mesh minimum size.
As the graph 1.1 show, the line getting linear when it approach 0.02 mesh min size. The more linear the graph, the more accurate result will be obtain. Besides then using the graph to determine the suitable mesh size, the suitability of mesh size can be determined by using the table. Table below shows the value from mesh 0.09mm min size with 0.19mm max size until 0.02mm min size with 0.05mm max size.
Mesh Min size (mm) Mesh max size (mm) Pressure difference (Pa)
0.09 0.19 167652.0
0.08 0.17 162693.6
0.07 0.15 183140.0
0.06 0.13 188118.7
0.05 0.11 191041.0
0.04 0.09 196625.1
0.03 0.07 197399.9
0.02 0.05 198272.5
Table 1.1: The mesh minimum size and mesh maximum size choose and the pressure calculated.
According to the table 1.1 shown, the value of the pressure difference keep fluctuate extremely between the value of 0.09mm min size with 0.19mm max size and 0.08mm min size with 0.17mm max size. The data difference between 2 values is around 5 thousand but as the mesh min size and mesh max size decrease, the difference between 2 values is reduced to 1 thousand which means the difference had been reduced thus the result will be more stable and the mesh size was determined. Other than that, the performance of the computer will be consider also because it was relate to efficiency of obtaining the result. Although, the smaller mesh size will result a better result but as the mesh size decrease, the computational time to obtain the result will be increase also. So, the most suitable mesh size will be determine by the performance of the computer and the value difference between the 2 continuous mesh sizes tested.
Section B: (Part 2)
Figure 2.1: Pressure applies on the body.
When a fluid through the sudden contraction of a pipe from a larger diameter to a smaller diameter, the streamlines cannot follow the sudden contraction of the geometry and hence converge gradually into the smaller diameter pipe which will results the changes in velocity and a pressure. (Azzopardi, 2011) When the fluid flow through a sharp or sudden contraction, the vena contraction is formed which the cross section area of the fluid in the smaller diameter pipe will reach the minimum and less than the diameter of the pipe. The majority of pressure losses will occurred as result of formed two circulation regions, which known as the back flow or flow separation, as shown in the diagram below. (NPTEL, 2003)
Figure 2.2: Setup of the layers of inflation.
In order to obtain a more accurate result, 20 layers of inflation has been added at the walls in the meshing step. This is due to the reason that the inflation can help to obtain more information of the fluid flow and observe the boundary condition of the reverse flow near the walls and contraction area.
In CFD simulation, 12 values of ratio of Lextend/D2 are picked from the range of 0.2 to 4.5 to as the value to avoid back flow at the pressure outlet fall in between these two values. As the ratio increases, it has no any obvious decreases in the back flow at the pressure outlet. Start from the ratio of 0.8, the back flow is noticeable disperse at the pressure outlet. Until the ratio reach 1.2, the back flow is completely does not exist at the pressure outlet. Hence, this can be concluded the reverse flow can be avoided at ratio of 1.2. The blue regions after the contraction represent the wake region created by the sudden contraction as shown in the diagrams below.
Figure 2.3: Pressure applies on the body with Ratio, Lectend/D2 =0.2
Figure 2.4: Pressure applies on the body with Ratio, Lectend/D2 =0.3
Figure 2.5: Pressure applies on the body with Ratio, Lectend/D2 =0.5
Figure 2.6: Pressure applies on the body with Ratio, Lectend/D2 =0.8
Figure 2.7: Pressure applies on the body with Ratio, Lectend/D2 =1.2
Figure 2.8: Pressure applies on the body with Ratio, Lectend/D2 =1.7
Figure 2.9: Pressure applies on the body with Ratio, Letend/D2 =2.0
Figure 2.10: Pressure applies on the body with Ratio, Lectend/D2 =2.2
Figure 2.11: Pressure applies on the body with Ratio, Lectend/D2 =2.8
Figure 2.12: Pressure applies on the body with Ratio, Lectend/D2 =3.5
Figure 2.13: Pressure applies on the body with Ratio, Lectend/D2 =4.0
Figure 2.14: Pressure applies on the body with Ratio, Lectend/D2 =4.5
Section B: (Part 3)
Figure 3.1: The Geometry of the question.
In this part, the pressure inlet (Pin) and pressure at sudden contraction (P1) are observed. The Pin and P1 were measured by taking the average pressure along y-axis. xy plot was used to get the data along the Pin and P1. To get the results of P1, a new surface (Line/Rake) is added to obtain the data of this line. The way to create a new surface is shown at figure below. However, only half of the results of P1 were taken. We only measure the average pressure from y = 0mm to y = 0.5mm. because the pressure observed starts becoming unstable above 0.5mm. The wall shear stress will make the pressure become lesser or unstable and will affect the overall result. Besides, we only need the information along the y-direction of the line, thus we set plot direction of x = 0 and y = 1. After that, the results were written to excel to get the average pressure of the line.
Figure 3.2: Create the new line and plot the graph with the line created.
L/D Inlet Pressure (Pa) Pressure at Contraction (Pa) Pressure Difference (Pa) Percentage Different
0.2 442051.000 198272.545 243778.455 –
0.3 450640.230 222166.500 228473.730 0.0628
0.5 402420.550 185398.417 217022.133 0.0501
0.8 332144.880 117891.542 214253.338 0.0128
1.2 303400.550 88183.531 215217.019 -0.0045
1.7 288407.970 73907.158 214500.812 0.0033
2 287316.690 72534.077 214782.613 -0.0013
2.2 284405.640 69014.185 215391.455 -0.0028
2.8 283665.080 69653.521 214011.559 0.0064
3.5 285810.156 74101.036 211709.120 0.0108
4 284689.290 70323.117 214366.173 -0.0126
4.5 286376.390 72709.931 213666.459 0.0033
Table 3.1: The Result of Calculation by different L/D.
From the table 3.1, the pressure at contraction is less than inlet pressure. This is because the flow of fluid through a contraction will cause the increase of velocity.
Graph 3.1: Graph of the pressure difference against L/D.
From the graph, the pressure difference is decreasing when the L/D is increasing. Besides, the rate of pressure difference is decreasing when the L/D is increase and until at L/D = 1.2, the pressure difference starts becoming stable. Besides, the percentage difference of L/D = 1.2 with 0.8 is 0.0045 (Three significant digits of precision). Thus, we can say that the pressure loss above L/D = 1.2 is almost constant.
Section B: (Part 4)
Graph 4.1: Graph of inlet pressure against L/D.
From this graph, the inlet pressure is high if the L/D is low. Besides, the inlet pressure will decrease if the L/D is increasing and the speed of inlet pressure decrease is gradually decrease if the L/D is increasing. After that, it will become constant when L/D is more than 1.2. Thus, we can say that the inlet pressure will be independent after the L/D = 1.2.
The pressure inlet will be higher when L/D is lower is because the loss in contraction is higher. This can be proved by the pressure loss in lower L/D is higher in previous graph in part iii. Thus, higher pressure in inlet is needed to overcome the pressure loss.
Thus, L/D = 1.2 is suggested to Shane. This is because when L/D = 1.2, the pressure loss and L extended are lowest. Based on the equation H_f= 4fl/D×v^2/g , can see that the head loss is directly proportional to the length of the pipe. Therefore, the lower the L, the lower the major loss. This is because straight pipe loss (head loss) will increase if the length of the pipe is longer. Thus, to get minimum straight pipe loss and pressure loss due to sudden contraction, L/D = 1.2 is the best choice for Shane.
Conclusion:
In conclusion, the objectives of doing this CFD assignment were achieved. Through this CFD assignment, the technique to perform Ansys Workbench was mastered. Besides, the pressure losses and velocity streamline in sudden contraction were studied. The higher the L/D ratio, the lower the pressure difference and the pressure start becoming constant after L/D = 1.2.
Through Workbench Ansys 16.0, the information that hardly obtained by human can be obtained because third order nonlinear differential equation is too complicated to be calculated and this equation can be solved by Ansys software. This equation is given by 2f^”’ (η)+f(η) f^” (η)=0.
Besides, the backflow will occur at the separation (corner at the sudden contraction) and wake region is produced. This is because the small pressure at the part which push back the liquid in opposite direction. Therefore, the larger the pressure difference, the more the back flow will occur. This can be proven by part three and we can see that the back flow is decreasing when the L/D ratio is increasing and start becoming stable because the pressure difference is start becoming stable too.