Similarity and Model Testing
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1 Similarity and Model Testing Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering
2 Modeling Model: A representation of a physical system that may be used to predict the behavior of the system in some desired respect Prototype: The physical system for which the predictions are to be made
3 Types of Similarity Three necessary conditions for complete similarity between a model and a prototype: 1) Geometric similarity ) Kinematic similarity 3) Dynamic similarity 3
4 1) Geometric Similarity A model and prototype are geometrically similar if and only if all body dimensions in all three coordinates have the same linear scale ratio. αα = LL mm LL or, λλ = LL LL mm All angles are preserved in geometric similarity. All flow directions are preserved. The orientation of model and prototype with respect to the surroundings must be identical. 4
5 ) Kinematic Similarity Kinematic similarity requires that the model and prototype have the same length scale and the same time scale ratio. One special case is incompressible frictionless flow with no free surface. These perfect-fluid flows are kinematically similar with independent length and time scales, and no additional parameters are necessary. 5
6 ) Kinematic Similarity Contd. Frictionless flows with a free are kinematically similar if their FFFF are equal: or FFrr mm = VV mm ggll mm = VV gggg = FFFF VV mm VV = LL mm LL 1 = αα In general, kinematic similarity depends on the achievement of dynamic similarity if viscosity, surface tension, or compressibility is important. 6
7 Example 1: FFFF Similarity 7
8 Example 1: FFFF Similarity Contd. For FFFF similarity, VV mm ggll mm = VV gggg VV mm VV = LL mm LL = αα1 = = Since QQ = VVVV, Also, Thus, QQ mm QQ = VV mmaa mm VVVV = VV mm VV AA mm AA = LL mm LL AA mm AA = αα1 AA mm AA = αα 5 QQ mm QQ = αα1 αα = αα 5 = 1 11 = 5 33, QQ mm = αα 5 QQ = = mm 33 /ss 5 8
9 3) Dynamic Similarity Dynamic similarity exists when the model and the prototype have the same length scale ratio (i.e., geometric similarity), time scale ratio (i.e., kinematic similarity), and force scale (or mass scale) ratio. To be ensure of identical force and pressure coefficients between model and prototype: 1. Compressible flow: RRRR and MMMM are equal. Incompressible flow: a. With no free surface: RRRR are equal b. With a free surface: RRRR and FFFF are equal c. If necessary, WWWW and CCCC are equal 9
10 Theory of Models Flow conditions for a model test are completely similar if all relevant dimensionless parameters have the same corresponding values for the model and then prototype. For prototype: Π 1 = φφ Π, Π 3,, Π nn For model: Π 1mm = φφ Π mm, Π 3mm,, Π nnmm Similarity requirement* Prediction equation Π mm = Π Π 3mm = Π 3 Π nnnn = Π nn Π 1 = Π 1mm *Also referred as the model design conditions or modeling laws. 10
11 Example : RRRR Similarity Drag measurements were taken for a 5-cm diameter sphere in water at 0 C to predict the drag force of a 1-m diameter balloon rising in air with standard temperature and pressure. Determine (a) the sphere velocity if the balloon was rising at 3 m/s and (b) the drag force of the balloon if the resulting sphere drag was 10 N. Assume the drag DD is a function of the diameter dd, the velocity VV, and the fluid density ρρ and kinematic viscosity νν. DD = ff dd, VV, ρρ, νν dd mm = 0.05 m dd = 1 m DD mm = 10 N at VV mm =? DD =? at VV = 3 m/s water model air prototype 11
12 Example : RRRR Similarity Contd. Dimensional analysis DD ρρvv dd = φφ VVVV νν Similarity requirement VVVV νν = VV mmdd mm νν mm Prediction equation DD ρρvv dd = DD mm ρρ mm VV mm dd mm 1
13 Example : RRRR Similarity Contd. (a) From the similarity requirement: or VV mm = VVVV νν = VV mmdd mm νν mm νν mm νν dd dd mm VV = m s m s 1 m 0.05 m 3 m s VV = 4.15 m s 13
14 Example : RRRR Similarity Contd. (b) From the prediction equation: or DD = DD ρρvv dd = ρρ ρρ mm VV VV mm DD mm ρρ mm VV mm dd mm dd dd mm DD mm = 1.3 kg m3 998 kg m 3 3 m s 4.15 m ss 1 m 0.05 m 10 N DD =.6 N 14
15 Example 3 Property Model Prototype Diameter, DD 8 in. 1 in. Angular velocity, ωω 40π rad/s 60π rad/s Flow rate, QQ? 6 ft 3 /s Fluid water water 15
16 Example 3 Contd. Dimensional analysis Δpp DD ωω ρρ QQ MMLL 1 TT LL TT 1 MMLL 3 LL 3 TT 1 rr = nn mm = 5 3 = mm = 3 repeating variables: DD for LL, ωω for TT, and ρρ for MM Π 1 = DD aa ωω bb ρρ cc Δpp = LL aa TT 1 bb MMLL 3 cc MMLL 1 TT = LL (aa 3cc 1) TT bb MM cc+1 = LL 0 TT 0 MM 0 Π 1 = DD ωω ρρ 1 Δpp = Δpp ρρωω DD 16
17 Example 3 Contd. Dimensional analysis contd. Π = DD aa ωω bb ρρ cc QQ = LL aa TT 1 bb MMLL 3 cc LL 3 TT 1 = LL (aa 3cc+3) TT bb 1 MM cc = LL 0 TT 0 MM 0 Π = DD 3 ωω 1 QQ = QQ ωωdd 3 Dimensionless eq. Δpp ρρωω DD = φφ QQ ωωdd 3 17
18 Example 3 Contd. Similarity requirement or QQ mm = QQ ωωdd 3 = ωω mm ωω QQ mm ωω mm DD mm 3 DD mm DD 3 QQ = 40ππ rad s 60ππ rad s 8 in. 1 in. 3 6 ft 3 s QQ mm = 1.19 ft 3 s 18
19 Example 3 Contd. Prediction equation Δpp ρρωω DD = Δpp mm ρρ mm ωω mm DD mm 5.5 or Δpp = ρρ ρρ mm ωω ωω mm DD DD mm Δpp mm 1.19 = 1 60ππ rad s 40ππ rad s 1 in. 8 in. 5.5 pppppp Δpp = 7.8 pppppp 19
20 Distorted Models It is not always possible to satisfy all the known similarity requirements. Models for which one or more similarity requirements are not satisfied are called distorted models. 0
21 Geometric similarity Model Testing in Water (with a free surface) LL mm LL = αα FFFF similarity VV mm ggll mm = VV gggg VV mm VV = LL mm LL 1/ = αα RRRR similarity VV mm LL mm νν mm = VVVV νν νν mm νν = LL mm LL VV mm VV = αα αα = αα3 1
22 Example 4
23 Example 4 Contd. For water at atmospheric pressure and at TT = 15.6 C, the prototype kinematic viscosity is νν = m /s. Required kinematic viscosity of model liquid: νν mm = νναα 3 = = m s Even liquid mercury has a kinematic viscosity of order 10-7 m /s still two orders of magnitude too large to satisfy the dynamic similarity. In addition, it would be too expensive and hazardous to use in this model test. 3
24 Model Testing in Water Contd. (with a free surface) Alternatively, by keeping νν mm = νν, RRRR similarity VV mm LL mm νν mm = VVVV νν VV mm VV = LL LL mm = αα 1 FFFF similarity VV mm gg mm LL mm = VV gggg gg mm gg = VV mm VV LL LL mm = αα αα 1 = αα 3 For example, for αα = 1/100, gg mm gg = Again, impossible to achieve. 3 =
25 Model Testing in Water Contd. (with a free surface) In practice, water is used for both the model and the prototype, and the RRRR similarity is unavoidably violated. The low-rrrr model data are used to estimate by extrapolation the desired high-rrrr prototype data. There is considerable uncertainty in using extrapolation, but no other practical alternative in model testing. 5
26 Model Testing in Water Contd. (with a free surface: Ship model testing) Assume: CC TT = ff RRRR, FFFF = CC ww FFFF + CC vv RRRR CC TT = Total resistance coefficient CC WW = Wave resistance coefficient CC vv = Viscous friction resistance coefficient Model Testing with the FFFF similarity CC ww = CC wwww = CC TTTT CC vvvv Extrapolation (with CC wwww = CCCC) CC TT = CC ww + CC vv = CC TTTT CC vvvv + CC vv CC vvvv RRee mm and CC vv RRRR obtained from flat plate data with same surface area of the model and the proto type, respectively. 6
27 Model Testing in Air Geometric similarity: LL mm LL = αα MMMM similarity: RRRR similarity: VV mm aa mm = VV aa VV mm VV = aa mm aa LL mm VV mm νν mm = LLLL νν νν mm νν = LL mm LL VV mm VV = αα aa mm aa Since the prototype is an air operation, need a model fluid of low viscosity and high speed of sound, e.g., hydrogen too expensive and danger. RRRR similarity is commonly violated in wind tunnel testing. 7
28 Example 5 8
29 Example 5 Contd. RRRR similarity: or VV mm = νν mm νν VV mm LL mm νν mm LL LL mm = VVVV νν VV = 10 νν mm νν VV since LL mm LL = 1/10. If νν mm νν = 1, Also, if VV = 55 mph, VV mm = 10VV VV mm = 550 mph Too large for simple tests. In addition, MMMM = 0.7 for the model and the compressibility effects become important while they are not for the prototype with MMMM = We will need νν mm νν < 1 for more realistic VV mm. 9
30 Example 5 Contd. νν variation with TT for air at standard atmospheric pressure (scaled with νν at 10 C): Thus, it would be better to have a colder wind tunnel. However, even with TT = -40 C, which gives νν mm νν = 0.707, the required VV mm = 10(0.707)(55) = 389 mph. 30
31 Model Testing in Air Contd. In practice, wind tunnel tests are performed at several speeds near the maximum operating speed, and then extrapolate the results to the fullscale Reynolds number. While drag coefficient CC DD is a strong function of Reynolds number at low values of RRRR, for flows over many objects (especially bluff objects), the flow is Reynolds number independent above some threshold value of RRRR. 31
32 Example 6 FF DD 1 ρρvv AA = φφ ρρρρρρ μμ WW = Width of the truck AA = Frontal area 3
33 Example 6 Contd. RRee mm mmmmmm = ρρ mmvv mm WW mm = kg m m m s μμ mm kg s/m = RRRR = ρρρρρρ = kg m m m s μμ kg s/m = Reynolds number independence is achieved for RRRR > , thus CC DD = FF DD 1 = CC ρρvv DDDD = 0.76 from the wind tunnel data AA FF DD = 1 ρρvv AA CC DD = kg m m s m 0.57m 0.76 = 3.4 kkn 33
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