Compressible Fluid Mechanics Project Assignment compressible fluid mechanics project this was due today so i need it ASAP. 411/511 Compressible Fluid Flow

Compressible Fluid Mechanics Project Assignment compressible fluid mechanics project this was due today so i need it ASAP. 411/511 Compressible Fluid Flow
Project 2: Wing Lift and Drag for Subsonic, Transonic and Supersonic Flight
Due date: Wednesday, April the 24th
Introduction and Background Idealization of inviscid flow over air vehicle is made by
neglecting viscous effects such as friction and separation of boundary layer. Thermal
conduction and diffusion are neglected too. Additionally, this assignment does not account
for finite span of the wing, that is, the flow is assumed 2-D, that is, the wing is assumed to
be infinitely wide. Finally, the flow past the wing is assumed to be steady.
In low-speed flight (commonly assumed that the free stream velocity has M?0.3 we can no longer assume that the flow is incompressible. Aerodynamic
problems can be classified according to whether the flow speed is below, near or above the
speed of sound. A problem is called subsonic (M? ranges from 0.3 to ~0.8) if all the speeds
in the problem are below than the speed of sound, and transonic if speeds both below and
above the speed of sound are present in the flowfield.
In general, equations of motion for inviscid compressible flow are non-linear partial
differential Euler equations. In many cases, analytic solutions are not possible. However it
is possible to linearize equations of fluid motion for subsonic and supersonic flows (but
not for transonic flows!) around thin wings and receive approximate analytical solution.
See Ch14 of the Textbook and Ref. [2] for details.
Airfoil performance is characterized by unit-less quantities such as the lift force
coefficient, CL , and drag force coefficient, C D (see Textbook, p. 187):
C L = L /(1 / 2 ?V 2 A) , C D = D /(1 / 2 ?V 2 A) .
Part 1 (analytical): Obtain lift and drag components of aerodynamic force for
supersonic airfoils.
Supersonic wings should provide a lift force (normal to the undisturbed flow)
accompanied by low drag force (in the direction of undisturbed flow).
Figure 1.1: Diamond-shape supersonic airfoil
The shape of airfoil to be used in subsonic flight is the well-known teardrop or
streamlined profile. In supersonic flight the wing design must be completely modified
owing to occurrence of detached shocks. If the leading edge of supersonic wing is not sharp
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(that is, rounded), the leading–edge shock wave will be detached and will cause large wave
drag. To minimize wave drag, the supersonic airfoil must have a pointed nose and also be
as thin as possible. The ideal case is a flat plate airfoil, possessing zero thickness. The flat
plate airfoil is, however, an idealization; structurally, such an airfoil is unsound.
One possible airfoil shape for supersonic flight is the diamond (double wedge)
profile shown in Figure 1. For positive angles of attack, flow over the lower surface is
turned through an oblique shock at A and then through an expansion fan at C. Flow over
upper surface for sufficient angle of attack is first expanded through a fan, centered at A,
and then is turned through another expansion fan at B. The difference in the pressure force
between lower and upper surfaces of wing creates lift force. We considered calculation of
lift and drag components of aerodynamic force for similar cases in class using tables and
charts pertinent to oblique shock waves and PM waves.
Another (easier) method of calculating lift and drag acting on supersonic wings is
based on linearized supersonic flow theory (see CH14 of Textbook). Basic assumption of
linearized supersonic flow theory is that pressure waves generated by thin airfoils are
sufficiently weak that they can be treated as Mach waves. Under this assumption, the flow
is isentropic everywhere. The pressure coefficients (including lift and drag coefficients)
obtained by linearized theory are available in Textbook, pp. 464-470. For example, lift
coefficient for the flat plate is obtained by linearized theory:
???? =
4??
,
(1.1)
2 ?1
????
where ? is the angle of attack and ??? is the supersonic flight Mach number.
For a symmetric double wedge airfoil, linearized theory gives the following
expression for drag coefficient [3] and Textbook, p. 470
???? =
4
2 ?1
????
?? 2
(?? 2 + (??) ),
(1.2)
where c is chord length and t/c is the relative thickness of the airfoil.
The linearized theory is an approximation valid only for small angles of attack and
thin airfoils. Otherwise, exact solution for flowfield, lift and drag components of force is
possible using analytic solutions for oblique shock wave (Ch6) and Prandtl-Meyer
expansion wave (Ch7).
This part of the project includes the following steps:
For the diamond-shape airfoil shown in Figure compute the lift and drag
coefficients for Mach number M=2.5. The angle of attack varies from -5 to 25 degrees.
The angle CAB is equal to 10 degrees. For this airfoil t/c=tan(5o).
1. Compute lift and drag coefficients as a function of angle of attack (take -5, 0, 5, 10,
15, 20 and 25 degrees) using analytic solution for oblique shock wave and Prandtl-
2
Meyer expansion wave. You can either adopt appropriate Tables and charts or write
your computer code (please attach it to your report). The latter method is more
efficient to avoid repetitive by-hand calculations.
2. Use linearized theory (Eqs (1-2)) to compute lift and drag coefficients as a function
of angle of attack.
3. Plot lift and drag coefficients as functions of angle of attack (obtained in #2). In the
same graphs plot lift and drag coefficients obtained in #1. Explain the difference
in lift coefficients obtained by two methods. How does the difference in lift force
depend on the angle of attack?
Part 2: Compute aerodynamic force exerted on airfoil in subsonic and transonic
flight using FLUENT/ANSYS CFD software
Transonic speed is an aeronautical engineering term referring to the condition of
flight in which the M? ranges of 0.8 to 1.2, i.e. 600–900 mph. When an aircraft moves at
the speed close to speed of sound, shock waves build up in front of it to form a single
shock wave. During transonic flight, the plane must pass through this large shock wave.
There are no analytical flow solutions available for transonic flows and therefore CFD
solutions and wind tunnel experiments are the options.
Figure 2.1: Example of transonic flowfield
Lift and drag calculations for compressible flow
Under the assumptions listed in the first section, the “inviscid” incompressible lift
coefficient increases linearly with ?:
C LINC = 2? (? ? ? o ) , (2.1)
where ? is the angle of attack in radians and ?o is the angle for which zero lift occurs for a
non-symmetrical airfoil. In your assignment, ?o =0o for symmetrical NACA0012 airfoil.
Instead of deriving entirely new equation for lift coefficient for inviscid subsonic
compressible flow, we can slightly change above equation for lift coefficient for
incompressible flow. Such adjustments are called compressibility corrections.
The widespread compressibility correction for subsonic flow is the Prandtl-Glauert
correction. This correction is obtained as a solution of Prandtl-Glauert equation which is a
linearized form of the full potential equation (see Textbook, pp. 462-463 and [2]). It stated
that the pressure coefficient Cp in a compressible flow can be derived from the pressure
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coefficient in an incompressible flow. Consequently the lift coefficient at flight at M? is
given by
C L = C LINC / 1 ? M ?2 (2.2)
The flow velocity is different on different positions on the wing. Assume we know the
Mach number MA of the flow over our wing at a given point A. The velocity of the flow on
top of our wing is generally bigger than the free stream velocity, V?. So we may have sonic
flow (M = 1) over our wing, while we are still flying at M? < 1. The critical Mach number Mcr is defined as the free stream Mach number M? at which sonic flow (M = 1) is first achieved on the airfoil surface (see Textbook, pp. 251-254). It is a very important value: when local Mach number gets bigger than unity, shock waves will appear (see Fig. 2.1 and Textbook, Fig 8.38). This causes additional drag. So the critical Mach number relates to the velocity at which the drag increases. Computational modeling of subsonic and transonic flight aerodynamics To generate geometry, grid, set-up boundary zones and computational parameters, you can use the .cas FLUENT file provided by the instructor through the website. To consider compressible flow with angle of attack of 4 degrees, please set the xcomponent of flow direction equal to 0.997564 and the y-component equal to 0.069756. This corresponds to angle of attack of 4 degrees. To set the CD and CL monitors correctly for the 4 degrees of angle of attack? For drag coefficient, put 0.069 for y icon and 0.997564 for x icon. For lift coefficient, swap x and y cos/sin that is x= -0.069 and y=0.997564 Set maximum number of iterations equal to 3000. Set stopping residuals equal to 10-3. Please conduct the following FLUENT 17 computations and present results in your report. 1. Model the 2-D inviscid flow about NACA0012 for Mach numbers M=0.6, 0.7, 0.75, 0.8, 0.85, 0.9, and 0.99. Consider the angle of attack equal to 4 degrees for all cases. For each case, present isolines (contours) of velocity magnitude, Mach number, and pressure. Notice for which flight Mach number the shock wave appears first and how its location at the airfoil surface changes with the Mach number. Present coefficients of lift and drag for each Mach number. Obtain the approximate value of critical Mach number. 2. Compare CL obtained by numerical modeling with those by Eq. (2). For which Mach numbers the Eq. (2) is useful and for which there is a significant difference between Eq. (2) and FLUENT result? 3. Obtain the values of C D as a function of Mach number. Comment about how the CD depends on the Mach number. References 4 1. Munson, B. R., Young, D. F. and Okiishi, T. H., Fundamentals of Fluid Mechanics, 6th edition, John Wiley & Sons. 2. Anderson, J.D., Fundamentals of Aerodynamics, McGraw-Hill. 3. M.A. Saad, Compressible Fluid Flow, Prentice Hall, NJ. 5 1. Does the Mach number need set for both the INLET and OUTLET boundary conditions, or just the INLET? Yes Both Inlet and Outlet BC need to set with the same Mach number 2. How to set-up Reference Values? (They are important to calculate correct values of lift and drag coefficients) For Reference Values there is a drop-down menu and you can ask to take reference values from INLET boundary conditions. As soon as you change Inlet BC, go again to the reference values and check that they (especially reference inlet velocity) do change. If not, change them again from the drop-down menu. 3. How to set the CD and CL monitors correctly, to account for the 4 degree angle of attack? For drag coefficient, put 0.069 for y icon and 0.997564 for x icon. For lift coefficient swap x and y (cos/sin), that is. x=-0.069 and y=0.997564 4. You told us to set the max number of iterations to 3000. Where do I set this? Press "Run calculation" on the left and select the number of iterations. 5. Is the Courant number okay at 5? Yes for implicit scheme 6. For new value of Mach number, should we initialize the flow field? There are two ways to iterate, both should give you similar values of flowfield, lift and drag. Using either way, you should change INLET and OUTLET boundary conditions first. Way 1. Initialize the flowfield: go to Solution Initialization and select Inlet from dropdown menu. Way 2. Change boundary conditions and continue to run the code. Purchase answer to see full attachment