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Recently Asked Gas Dynamics Questions

Expert help when you need it
  • Q1:4. Download the Matlab code posted to the Canvas (in supplementary materials) for integrating the problem of a compressible laminar boundary layer over a flat plate with zero streamwise gradients in the overriding freestream and with the viscosity following a temperature power law with exponent n = 0.7. The gas is calorically perfect and assume that the edge conditions involve a Mach number Me = 5. Use and modify this code accordingly to answer the following questions: a) Determine the ratio of the adiabatic wall temperature Ta,w to the edge temperature Te, along with the recovery factor r. State whether the approximation r = √Pr is a good one for these conditions. (10 points) b) For an isothermal wall at Tw/Te = 3, plot the dimensionless profiles of the static temperature T/T, stagnation temperature To/Te, density p/p and streamwise velocity u/ve, all as a function of the normalized wall-normal coordinate y √Reex/x, where x is the streamwise coordinate and Ree,x is the edge Reynolds number based on x. (30 points) c) For the conditions described in (b), calculate the products Cr√ Reex and CH√ Reex, along with the Reynolds analogy factor 2CH/C, where Cy is the skin friction coefficient and Cy is the Stanton number. Compare the Reynolds analogy factor with Pr-2/3. (10 points) Attach a screenshot of your modified version of the "main.m" script with your solutions.See Answer
  • Q2:3. The Blasius solution to the steady-state two-dimensional incompressible laminar boundary layer equation for uniform flow over a flat surface can be shown to be as follows: 4(x,y) = √vUxf(n) Where x and y are streamwise and wall-normal coordinates, respectively, v is the kinematic viscosity (which is a constant for an incompressible flow), U., is the freestream velocity, 4' is the stream function (which is related to velocity as u = 4/ǝy and v = -ǝч/ǝx in an incompressible flow), u and v are the component of the velocity in x and y directions, respectively, n = is the similarity variable, and f can be found by solving the following y √vx/U∞ ordinary differential equation (ODE): + Boundary conditions: f(0) = f'(0) = 0 and f'(∞) → 1 The solution to this ODE has the following asymptotic forms for small and large values of the similarity variable: {f(n) = 1² f(n) =n-√3 6 for n << 1 for >> 1 a) Using the above results, obtain an expression for the nondimensional surface shear stress, i.e., Tw, in terms of the Reynolds number based on the distance x. (5 points) b) Find the nondimensional vertical component of velocity at y→ ∞, i.e., " boundary layer in terms of the Reynolds number based on the distance x. (5 points) う in theSee Answer
  • Q3:1. A wedge with semi-angle = 20° is moving through air (T = 200 K, Pr = 0.7225 and y = 1.4) at a Mach number more than one. The wedge is kept at temperature Twall = 1600 K. Calculate the free-stream Mach number, Ma.., at which the wedge wall temperature becomes equal to the adiabatic wall temperature of the boundary layer. What is the corresponding Mach number at the edge of the boundary layer, Maę? Hint: Assume that the recovery factor is equal to √Pr, the gas is calorically perfect, and ignore the effect of boundary layer on the inviscid flow. (30 points) Ma∞ > 1 Tw B 20 shock 20B boundary-layer edgé u(y) Mae, Pes Tes HeSee Answer
  • Q4:/n Assignment 4: Supersonic Flow over a Cone ASEN 5151 Spring 2024 For this assignment you should upload your submission in two parts: • Any written work for each part, along with any code outputs (tables or graphs - includ- ing any requested numerical outputs) compiled into a single PDF document should be uploaded to the "Assignment 4" assignment on Gradescope. • All code used to generate your results should be uploaded to the "Assignment 4 - Code" assignment on Gradescope. If either part is missing, you will receive a zero for the entire assignment. Write a code to solve the conical shock wave problem in which you specify the cone angle Sc and freestream Mach number M₁ and calculate the shock wave angle, 0. The inputs to your program should be the specific heat ratio y, the freestream Mach number M₁, the cone angle dc, the convergence tolerance for the cone angle ɛ, and the number of integration points N to be used. The outputs to your code should consist of the shock wave angle 0s, the stagnation pressure ratio across the shock p02/P01, the cone surface Mach number Mc, the cone surface pressure coefficient (Cp)c, and at each angular location, the non-dimensionalized velocity components V, and V2, the Mach number M, and the local-to-freestream ratios of static pressure, static temperatures, and static density, p/p₁, T/T₁, and p/p₁. To check for convergence of the cone angle, compute the relative error and compare with the convergence tolerance ε: r و - relative error || Sc — § (²) | |8c| To demonstrate your program, run it for a freestream Mach number of M₁ = = 2.4, a cone - (8, 16, 32) for air (y = 1.405) semi-angle of c = 16°, using three different grid spacings: N with a cone angle convergence criteria of € = 10-6. For each grid spacing, please output the following to at least 5 significant figures: (a) The shock wave angle; (b) The cone surface pressure coefficient; (c) The cone surface Mach number. Compare to the NACA 1135 charts. In addition, include plots of the Mach number and pressure distribution (P/P₁) between the shock and cone surface. For your x-axis, plot the coordinate as descending (such that from left-to-right you march from the shock to the cone surface). For both plots, include all three grid solutions on one plot to verify grid convergence. Finally, export a .dat file with the angular distribution of , Vr, V₁, M, P/P₁, and T/T₁ for the N - 16 grid reported to 5 significant figures. Include this table in your PDF write-up. = 1See Answer
  • Q5: ARO 3111 Computer Assignment Part I Conical Flow: Supersonic Flow Past Un-Yawed Cone Objective: Develop a computer program in a high-level computer language (MATLab, C++, etc.) to numerically calculate the properties of the conical flow generated by a right circular cone flying at a supersonic Mach number at zero pitch and zero yaw through an inviscid perfect gas. The cone has a sharp tip (vertex). Numerical Approach: To accomplish this, numerically integrate the Taylor-Maccoll (T-M) equation, which is a second order and very nonlinear ODE. Do so by first converting this second order equation into two first-order ODEs. Solve these two by the use of the powerful and popular algorithm: 4th-order (Classical) Runge-Kutta method. Use the "inverse" approach described in Anderson (Modern Compressible Flow, 2021, Section 10.4, pp. 373-374) by starting at the given conical shock wave angle 6, and marching in the negative -direction, in small increments of A from the shock towards the cone. O is the spherical angular variable, measured from the cone axis up to an arbitrary ray emanating from the cone vertex. Recall that "conical flow" is just the flow between the shock and the cone body, is fully isentropic, and is characterized by the remarkable characteristic that all flow properties are constant along each ray, though they vary from ray to ray. Of course, the free stream flow is isentropic ahead of the shock. The only region of non-isentropic flow is across the conical shock wave, which like all other shocks is adiabatic. Thus, (using Anderson's notation) T = constant everywhere, but po will drop across the shock, and will remain constant behind the shock. Also, note that, although, the shock wave is 3-D, the shock is locally planar, and thus locally can be treated by the use of 2-D oblique shock theory. Governing Equations: In Anderson (2021), T-M equation is Eq. 10.13, but instead work with the more convenient non-dimensional version of it in Eq. 10.15. You will also be using Eq. 10.14 to find the 0-component of velocity V. The latter will look the same in non-dimensional form, except for adding a prime on each side. Once you have found the two components of velocity, V, and V, you can combine them to find the magnitude of the local velocity vector V. From the latter, you can now find the local Mach number M from Eq. 10.16, which you will need for isentropic flow calculations in the conical flow field. For convenience, drop the primes from these three equations, i.e., Eqs. 10.14, 10.15 and 10.16. Notice a small misprint in Anderson p. 374, Fig 10.4: V-component of velocity just downstream of the shock should be parallel to and just downstream of the shock, not as shown. Notation: Use that of Anderson (2021): • Station 1 or ∞ is the free stream 8 • Station 2 is right behind the shock • Station c is at the surface of the cone • Subscript o is stagnation values • Cone angle is • Conical shock angle is ẞ or • Initial turning of the streamline as the flow crosses the shock from 1 to 2 is 8 which is <0 Case Study: 1. Run your program for: M₁ = 6 y=1.4 2 0. = 12° A0=-0.1° (this is small enough to give you good accuracy, we are marching in the negative direction, thus a negative sign on the step size). Find that produces this shock wave and the corresponding M. 2. Once your program is working, we will calculate and extract more data from it: calculate non-dimensional flow properties on each ray and plot as a function of Programming: See the attachment for the governing equations and the 4th-order Runge-Kutta (R-K) algorithm. In your computer program, use all the same notations as in the attachment. Structure your computer program with separate modules and each one should start with one or more comment statement(s) that describe what is being done in that module and also define your notation for the module. Note that you will need the free stream velocity nondimensionalized by Vmax, which is obtained from Eq. 10.16, applied to the free stream. You will see that the answer is not 1, but less than 1. You will need this value to compute your initial conditions at Station 2 to start your R-K computations. Hint: The final answers for 0, Cpc, and M can be read from Charts 5, 6, and 7 in NACA 1135, good checks on the veracity of the core of your program. 3 ARO 3111 Computer Assignment Part II Conical Flow: Supersonic Flow Past Un-Yawed Cone Now that your computer program is working, we will add some gas dynamics to it and extract some of the flow properties in the conical flow region, between the shock and the cone. Again, our Case Study is: M₁ = 6 y=1.4 0. = 12° A0=-0.1° You already found the values of 0 and M. for this flow from your two plots. Now, continue your work as follows: 1. Use locally planar oblique shock relations to obtain from flow properties at Station 1 (the free stream), flow properties at Station 2 (right behind the shock), as you did in Problem 10.1 of Anderson (2021) in PS9. Do your work in non-dimensional form, by hand calculation, with 4 significant figures, as well as by incorporating into your program, to find: M2, P2 P1, P2 P₁, T₁/T₁. They should agree completely. This also helps debug the early part of your program. 2. Now, recalling that flow from Station 2 to Station c is isentropic, use your previously developed program to compute and plot, in one color graph, local values of these properties as a function of the spherical variable 0, i.e., M, p/p₁, p/p₁, T/T. Theta will now go from the above to O, that is, an inverted scale. Add the results in (1) to complete your plot for theta between and 45°, to fully see what happens to the fluid particle as it approaches the cone. 3. Add one table of all three plots vs 0 in increments of one degree. 4. You will present your work in a report.See Answer
  • Q6:4.13 The total flow at a wastewater treatment plant is 600 m³/day. Two biological aeration basins are used to remove BOD from the wastewater and are operated in parallel. They each have a volume of 25,000 L. In hours, what is the aeration period of each tank?See Answer
  • Q7:4.12 Calculate the hydraulic residence times (the retention time) for Lake Superior and for Lake Erie using data in Table 4.3.See Answer
  • Q8:pona, units of mg/L? 4.2 A mixture of two gas flows is used to calibrate an air pollution measurement instrument. The calibra- tion system is shown in Figure 4.23. If the calibration gas concentration Ceal is 4.90 ppm,, the calibration gas flow rate Qcal is 0.010 L/min, and the total gas flow rate Qtotal is 1.000 L/min, what is the concentration of calibration gas after mixing (Ca)? Assume the con- centration upstream of the mixing point is zero.See Answer
  • Q9:4.1 A waste stabilization pond is used to treat a dilute municipal wastewater before the liquid is dis- charged into a river. The inflow to the pond has a flow rate of Q = 4,000 m³/day and a BOD concentration of Cin=25 mg/L. The volume of the pond is 20,000 m³. The purpose of the pond is to allow time for the decay of BOD to occur before discharge into the environ- ment. BOD decays in the pond with a first-order rate constant equal to 0.25/day. What is the BOD concen- tration at the outflow of the pond, in units of mg/L?See Answer
  • Q10:Q1 (a) Define, in words, the zeroth law of thermodynamics. (b) Explain, in words, the concept of entropy and its connection to the second law of thermodynamics. (c) Identify and describe briefly four forms of heat transfer [2/10 marks] (e) Define, in words, the third law of thermodynamics. [2/10 marks] [2/10 marks] (d) Give two reasons why the maximum feasible officiency of a cyclic heat power plant cannot be achieved in practice. [2/10 marks] [2/10 marks]See Answer
  • Q11:M Consider a 2D inviscid flow supersonic wind tunnel as in Fig. 1.4 Nozzle Mtest Test section Fig. 1 Wind Tunnel Mexit L Design a wind tunnel nozzle that maintains uniform supersonic flow in the test section at a Mach number of 3.0, given an inflow Mach number of 0.3. Provide a detailed explanation of the designed geometry, including the contour results.See Answer
  • Q12: PROBLEM 2 Problem 3.19. Consider the roof of a car that is moving in still, atmospheric air with a speed of 100 km/h. The air temperature is 300 K. a) Assuming that the car's roof is adiabatic, calculate the temperature of the roof's surface temperature at 0.25 m behind the leading edge of the roof. b) Assume that a bug, which can be idealized as a sphere with 0.29 mm diameter, is trapped in the boundary layer at the location described in part a so that its center is 1.13 mm away from the wall. Estimate the drag force experienced by the bug. Also estimate the velocity difference across the bug's body. You can find the drag coefficient for the bug from C₁ = [√25/Re, +0.5407], where d is the diameter of the bug. c) How would you find the air temperature where the bug is located? (Note that you do not need to do calculations. You only need to explain.)See Answer
  • Q13: Problem 3.21. Microscopic particles that are suspended in gas are driven from high temperature to low temperature regions. This process is called thermophoresis. In the absence of other particle diffusive transport mechanisms, the slip velocity (velocity between gas and particle) caused by thermophoresis can be found from [Talbot et al., 1980; Friedlander, 2000]: UTP = mol k (17+ C,(2) k₁ P -2C,V = V k (1+6C Kn) 1+2 +4C, Kn where d is the particle diameter, kp is the thermal conductivity of the particle, all properties without a subscript represent the gas, and C, 1.17; C = 2.18; C = 1.14. Kn₁ = 2/d πΜ 2R T u VT T +C, (2Kn) C- 1/2 P (Knudsen number) (1.5.10) (Gas molecular mean free path) C=1+2Kn [1.257 +0.4cxp(-0.55/ Kn)] (Cunningham correction factor) Consider a flat and horizontal surface that is at a temperature of 398 K, and is cooled by a parallel air flow. The air has a pressure of 0.1 bar and a temperature of 253 K, and flows with a far-field velocity of 20 m/s with respect to the surface. At a distance of 0.5 m downstream from the leading edge of the surface, calculate the thermophoretic velocity in the vertical (y) direction of a metallic spherical particle that is 0.5 µm in diameter and has the thermophysical properties of cupper, when it is 1 mm away from the surface. How does this velocity compare with the fluid velocity in the y direction?See Answer
  • Q14:6. It was shown that, during starting, an isentropic diffuser would experience a detached shock and consequent losses. In order to swallow the shock, a fixed- geometry diffuser must be overspeeded. However, as shown in Fig. 6.9, as the design Mach number increases, the required overspeeding increases very rapidly, so that even if the aircraft could be infinitely overspeeded, the design Mach number would be limited to a finite value. Assuming one-dimensional flow and constant y (1.4), determine the absolute maximum design Mach number for which an otherwise isentropic diffuser of fixed geometry may be expected to start, any amount of overspeed being possible.See Answer
  • Q15:5. Sketched are three supersonic inlets: an isentropic inlet, the Kantrowitz- Donaldson inlet of Fig. 6.10, and a simple normal shock inlet. For flight Mach numbers M from 1 to 4, calculate the plot poz/Po as a function of M. with each inlet operating with best back pressure.See Answer
  • Q16:4. The figure indicates a two-dimensional diffuser that produces no net turning of the flow. For the geometry shown, calculate the overall stagnation pres- sure ratio for a flight Mach number of 3.0. Neglect all losses except those occurring in the shocks. Would this diffuser be easy to start?See Answer
  • Q17:3. The figure indicates a hypothetical one-dimensional supersonic inlet installed in a wind tunnel and equipped with a throttle valve by which the downstream static pressure p: might be varied. Suppose that the inlet is designed for a Mach number M-3.0 and that with this flight Mach number the shock has been swallowed and an internal shock exists, as at. Neglecting all losses except those occurring in the shock, calculate and plot the shock Mach num- ber M, and the stagnation pressure ratio Puz/pa. as a function of the static pressure ratio p/p. (for y=1.4). Let p/p. range from unity to well beyond that value which disgorges the shock. See Answer
  • Q18: Page 4) Given: Steady-state-steady-flow of a compressible gas. A thin normal shock occurs and is shown for a control volume fixed on the wave. Relative flow at (1) approaches the shock from upstream and uniform relative flow at (2) movies away downstream of the shock. For this flow situation only the pressures and densities are known. The gas is not thermally perfect. Starting from 1st principles develop the following general relationship for the shock speed of a wave moving into a static fluid. This is the shock speed (left-right) viewed from the ground.See Answer
  • Q19:Problem 2.) SELECT THE BEST RESPONSE Given: These equations are valid for 1-D SSSF with no external heat transfer and no external work for a calorically perfect gas with constant R, Cv, Cp & y. CIRCLE the best response (A, B, C or D) concerning the validity of each equation for other types of matter.See Answer
  • Q20:) Prob 1.) Consider a quasi-1-D steady adiabatic flow of 100 kg/s of neon gas (a monatomic gas that is calorically perfect) confined in a converging-diverging nozzle. A normal shock occurs at the nozzle exit plane at (2)→ (3) as shown. Friction is insignificant.See Answer
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