Fluids Project #1: Hydrostatics This project is designed to introduce you to aspects to hydrostatic problems: dams and standard atmospheres. You will need to use spreadsheets and/or MATLAB to solve these problems. The project has multiple parts as shown below: Topic Points Earth Standard Atm 20 Alt. Standard Atm 20 Dams 80 Date Assigned 2/19 Date Due 3/1 When Part A approved 2/19 3/8 3/8 You may use the sample spreadsheets on Canvas as your starting point. The project should be completed in groups of 2-3 students. Part A: Standard Atmosphere 1. (10 points) Using the formulas for the standard atmosphere developed in class to complete a spreadsheet to compute the temperature, pressure, density for each kilometer of altitude (h) between 0 and 100 km. Put these values in columns C, D, and E as indicated. Indicate either the constant temperature in K or lapse rate in K/km in column B. 2. (10 points) Construct a parallel set of columns in USCS units (feet, OR, psf, slug/ft³) in columns G, H, I, and J by converting the appropriate values from columns A, C, D, and E 3. This result must be submitted, checked, and found correct before Part B is assigned for your team. Part B: Alternate Standard Atmosphere (20 points) 4. Upon successful completion of Part A, your team will be provided data for a different atmosphere - either Earth at different latitudes or data for the atmosphere of another planet or moon. You will need to construct the best standard atmosphere possible for your data. The number of segments (constant or linear temperature) allowable will depend on the atmosphere – for Earth, the maximum will be eight segments. 5. The data will be in the form of pressure vs. temperature. You will need to determine the altitude and density for each data point and construct four columns in a new spreadsheet as with the standard atmosphere from part A (you can use columns G-J but stick to metric units). For your standard atmosphere fit, construct columns as per A-E in part A. - 6. Now for the tricky part – what is the best standard atmosphere? We will use a R² value to determine the fit. Our goal, however, is not to evaluate the temperature fit, but the density fit. So, you will have to determine the density given by your standard atmosphere for each data point provided for the real atmosphere. Then, at each point, compute the following: = A log(p) log10(Pfit) - log10(Pdata) Put this result in another column. You should have as many values as data points in the original data (n points). Then calculate the R² value by summing the following over all the data points: == n. n Σ10810 (Paata): i=1 n n Stot = Σ(10910(Paata): – Α)2 - Serr = Σ (Alog(p);)² i=1 i=1 Then, the R² value is just: R² = 1. Serr Stot This value is designed to evaluate the accuracy of simple regression fits. We use the log of the density to avoid issues of rapidly changing magnitudes (since density falls off exponentially). Essentially, this approach drives one towards a "least-squared fit” in order to minimize R². So, compute this value, and then continue to redesign your standard atmosphere to try to get this value as close as possible to 1. Once you have your best result, submit this second spreadsheet providing details of your error calculations as well as the final values in your spreadsheet in a clearly labeled fashion (you only have to provide details and calculations for your final version of the standard atmosphere as that is what I am going to evaluate). Part C: Numerical Integration of Hydrostatic Forces (80 pts) From solving the computation of hydrostatic forces on a curved dam, it is apparent that it is possible to determine the hydraulic force and center of pressure as follows: Fx = −√√ pgh¸ (î · ñ)dA ≈ –Σ pgh¸¸n (î · î„)ªµ‚ A n n n' F₁ =- pgh(n)dA = −Σ pghc¸„(ĵ · ñ„)ªµ == y x' = −. A 1 1 ·√√ pghx(j · ñ)dA y A SS pgh¸y(î ñ)dA y' = Fr x A n ≈1 n -Σpghc,nxn (j•^n)An F₂ -Σpghcny' (in) An F x n Note I am using he here for the centroid depth. While the integral is the pure approach, we typically approximated integral with a summation. However, since integration is just a fancy way to say addition, if we split the sum into sufficiently tiny areas, a good approximation of the integral can be achieved. If I know the shape of the gate, then a simple means of splitting the gate into areas is dividing it up into columns of fluid as show in the figure below. Now assume that each column has a sloped, rectangular bottom. If we chose enough columns, this is a reasonable approximation of the curved surface. At this point the centroid, center of pressure depth, area, and unit normal are easy to define for each column. Thus, the above sums can be taken and the desired result obtained. For example, consider a gate shaped like a quarter circle with a 1 m radius with the top of the water located 1 m above the top of the gate and 1 m in width (normal to page). The gate has been divided into four columns of equal width. Relative to the local coordinate system, the locations of the column endpoints on the gates are: x 1 m Pa 1 m Column endpoints y 0 2 1 m Pa 0.25 1.968246 0.5 1.866025 0.75 1.661438 x²+(y−1)²=1 1 To determine the necessary properties, treat the gate as series of four rectangular gate sections at different angles. The centroid depth is simply the depth of the midpoint between the two endpoints. The area of each section is defined by the width times the length of the gate side: x 1 m Pa 1 m Column endpoints y Segment properties h centroid length area 1 m 2 1.984123 0.252009 0.252009 Pa 0.25 1.968246 1.917136 0.270091 0.270091 0.5 1.866025 1.763732 0.323042 0.323042 0.75 1.661438 1.330719 0.707107 0.707107 x²+(y-1)²=1 1 1 The angle 0 relative to the horizontal is determinable from trigonometry. The surface unit normal pointing into the fluid is then given by: n = − sin o î - cos 0 ĵ X Column endpoints Segment properties Z h centroid length 2 0.25 1.968246 area tangent theta (rad) i 0.12634 1.984123 0.252009 0.252009 0.127017 -0.126 -0.99203 1.917136 0.270091 0.270091 0.408882 0.38814 -0.37847 -0.92561 0.5 1.866025 1.763732 0.323042 0.323042 0.81835 0.68583 -0.63332 -0.77389 0.75 1.661438 1.330719 0.707107 0.707107 2.645751 1.209429 -0.93541 -0.35355 1 1 noting that conventionally we define z as positive downward for these types of hydrostatics problems. The height of the center of pressure can come from the baseline formula for an angled rectangle gate: y' = yc + I sin² (0) Уса =yc + 112 bh³ sin² (0) y.bh =yc + h² sin² (0) 12yc Since the surface is flat, x' must be located on the gate at y'. This can be solved by keeping the tangent constant (a technique known as similar triangles back when I took geometry): Xnyn x' Xc,nyc,n Xn+1Yn+1 Column endpoints Segment properties X y 0 h centroid length 2 0.25 1.968246 y' X' 1.984123 0.252009 1.984165 0.124667 1.917136 0.270091 1.91759 0.373889 0.5 1.866025 1.763732 0.323042 1.765709 0.622583 0.75 1.661438 1.330719 0.707107 1.358116 0.864645 1 1 Having solved these properties, it is now straightforward to complete the above sums to generate the values for the entire gate: Column endpoints X Segment properties y h centroid i j y' X' dFx 0 2 0.25 1.968246 1.984123 1.917136 -0.37847 -0.92561 -0.126 -0.99203 dFy 1.984165 0.124667 618.0708 4866.061 606.6352 1226.355 1.91759 0.373889 1922.47 4701.775 1757.943 3686.509 dx' dy' 0.5 1.866025 1.763732 -0.63332 -0.77389 1.765709 0.622583 3539.817 4325.552 2693.017 6250.287 0.75 1.661438 1.330719 -0.93541 -0.35355 1.358116 0.864645 8634.643 3263.588 2821.844 11726.85 1 1 Results 14715 17156.98 0.459256 1.555556 This approach is comparable to the trapezoidal rule for numerical integration. Your assignment is to construct a series of spreadsheets similar to that above and the example provided on the website. For these problems assume that g = 9.81 m/s² and p = 1000 kg/m³. 1. (30 points) For the attached dam arrangements, construct spreadsheets similar to the one available on the website to determine the hydraulic force vector and the location of the center of pressure. For each surface compute solutions using 4 columns, 8 columns, 16 columns, and 32 columns. Assume each of the gates has a width of 1 m normal to the page. 2. (15 points) The concept of grid independence is critical to computational solutions. Grid independence in this problem means that there are sufficient columns such that doubling the number of columns does not significantly change the answer. Compare your results in all four variables (Fx, Fy, x', y'). If the change is less than 0.5% in all the variables when the columns are doubled, these results we will call grid independent. For each dam arrangement, determine at which step (4 to 8, 8 to 16, 16 to 32, not yet grid independent) grid independence is achieved. Which quantities do not change during this process? 3. (15 points) Check your results for the 4 cases by calculating the center of pressure and gage pressure values through direct integration of the dam profiles. 4. (20 points) Your team is performing consulting work for a company planning a new dam on the Mississippi River. They wish to see your handiwork on real dams. Blueprints for several dams will be placed on Canvas. You will need to compute the total hydraulic force and center of pressure location on two of these dams as instructed for each case. Submit the needed information about the dam (what is it, geometry, where you found the information, any assumptions you made), your spreadsheet calculations, and your resultant answer. Include the data requested on summary answer sheet. For these problems, your group must submit the spreadsheets and the completed summary answer sheet, along with the additional requested materials for part 3.

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Fluids Project #1: Hydrostatics This project is designed to introduce you to aspects to hydrostatic problems: dams and standard atmospheres. You will need to use spreadsheets and/or MATLAB to solve these problems. The project has multiple parts as shown below: Topic Points Earth Standard Atm 20 Alt. Standard Atm 20 Da