5. Dimensional Analysis. 5.1 Dimensions and units

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1 5. Diensional Analysis In engineering the alication of fluid echanics in designs ake uch of the use of eirical results fro a lot of exerients. This data is often difficult to resent in a readable for. Even fro grahs it ay be difficult to interret. Diensional analysis rovides a strategy for choosing relevant data and how it should be resented. This is a useful technique in all exerientally based areas of engineering. If it is ossible to identify the factors involved in a hysical situation, diensional analysis can for a relationshi between the. The resulting exressions ay not at first sight aear rigorous but these qualitative results converted to quantitative fors can be used to obtain any unknown factors fro exeriental analysis. 5. Diensions and units Any hysical situation can be described by certain failiar roerties e.g. length, velocity, area, volue, acceleration etc. These are all known as diensions. Of course diensions are of no use without a agnitude being attached. We ust know ore than that soething has a length. It ust also have a standardised unit - such as a eter, a foot, a yard etc. Diensions are roerties which can be easured. Units are the standard eleents we use to quantify these diensions. In diensional analysis we are only concerned with the nature of the diension i.e. its quality not its quantity. The following coon abbreviation are used: length ass tie force teerature M T F Θ In this odule we are only concerned with, M, T and F (not Θ). We can reresent all the hysical roerties we are interested in with, T and one of M or F (F can be reresented by a cobination of TM). These notes will always use the TM cobination. CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity

2 The following table (taken fro earlier in the course) lists diensions of soe coon hysical quantities: Quantity SI Unit Diension velocity /s s - T - acceleration /s s - T - force energy (or work) ower N kg /s kg s- M T - Joule J N, kg /s kg s - Watt W N /s kg /s ressure ( or stress) Pascal P, N/, kg//s - Ns M T - kg s - M T - - N kg - s - M- T - density kg/ kg - M - secific weight N/ relative density kg/ /s kg - s - M - T - a ratio no units viscosity N s/ surface tension kg/ s N/ kg /s 5. Diensional Hoogeneity N s - no diension kg - s - M - T - - N kg s - MT - Any equation describing a hysical situation will only be true if both sides have the sae diensions. That is it ust be diensionally hoogenous. For exale the equation which gives for over a rectangular weir (derived earlier in this odule) is, Q / B gh The SI units of the left hand side are s -. The units of the right hand side ust be the sae. Writing the equation with only the SI units gives CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity

3 ( ) / / s s s i.e. the units are consistent. To be ore strict, it is the diensions which ust be consistent (any set of units can be used and sily converted using a constant). Writing the equation again in ters of diensions, ( ) / / T T T Notice how the owers of the individual diensions are equal, (for they are both, for T both -). This roerty of diensional hoogeneity can be useful for:. Checking units of equations;. Converting between two sets of units;. Defining diensionless relationshis (see below). 5. esults of diensional analysis The result of erforing diensional analysis on a hysical roble is a single equation. This equation relates all of the hysical factors involved to one another. This is robably best seen in an exale. If we want to find the force on a roeller blade we ust first decide what ight influence this force. It would be reasonable to assue that the force, F, deends on the following hysical roerties: diaeter, d forward velocity of the roeller (velocity of the lane), u fluid density, revolutions er second, N fluid viscosity, Before we do any analysis we can write this equation: or F φ ( d, u,, N, ) 0 φ ( F, d, u,, N, ) where φ and φ are unknown functions. These can be exanded into an infinite series which can itself be reduced to F Κ d u θ N r σ where K is soe constant and,, q, r, s are unknown constant owers. Fro diensional analysis we CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 4

4 . obtain these owers. for the variables into several diensionless grous The value of K or the functions φ and φ ust be deterined fro exerient. The knowledge of the diensionless grous often hels in deciding what exeriental easureents should be taken. 5.4 Buckingha s π theores Although there are other ethods of erforing diensional analysis, (notably the indicial ethod) the ethod based on the Buckingha π theores gives a good generalised strategy for obtaining a solution. This will be outlined below. There are two theores accredited to Buckingha, and know as his π theores. st π theore: A relationshi between variables (hysical roerties such as velocity, density etc.) can be exressed as a relationshi between -n non-diensional grous of variables (called π grous), where n is the nuber of fundaental diensions (such as ass, length and tie) required to exress the variables. So if a hysical roble can be exressed: φ ( Q, Q, Q,, Q ) 0 then, according to the above theore, this can also be exressed φ ( π, π, π,, Q -n ) 0 In fluids, we can norally take n (corresonding to M,, T). nd π theore Each π grou is a function of n governing or reeating variables lus one of the reaining variables. 5.5 Choice of reeating variables eeating variables are those which we think will aear in all or ost of the π grous, and are a influence in the roble. Before coencing analysis of a roble one ust choose the reeating variables. There is considerable freedo allowed in the choice. Soe rules which should be followed are i. Fro the nd theore there can be n ( ) reeating variables. ii. When cobined, these reeating variables variable ust contain all of diensions (M,, T) (That is not to say that each ust contain M, and T). iii. A cobination of the reeating variables ust not for a diensionless grou. iv. The reeating variables do not have to aear in all π grous. v. The reeating variables should be chosen to be easurable in an exeriental investigation. They should be of ajor interest to the designer. For exale, ie diaeter (diension ) is ore useful and easurable than roughness height (also diension ). CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 5

5 In fluids it is usually ossible to take, u and d as the threee reeating variables. This freedo of choice results in there being any different π grous which can be fored - and all are valid. There is not really a wrong choice. 5.6 An exale Taking the exale discussed above of force F induced on a roeller blade, we have the equation 0 φ ( F, d, u,, N, ) n and 6 There are - n π grous, so φ ( π, π, π ) 0 The choice of, u, d as the reeating variables satisfies the criteria above. They are easurable, good design araeters and, in cobination, contain all the diension M, and T. We can now for the three grous according to the nd theore, π a u b d c F π a u b d c N π a u b d c As the π grous are all diensionless i.e. they have diensions M 0 0 T 0 diensional hoogeneity to equate the diensions for each π grou. we can use the rincile of For the first π grou, π a u b d c F a b c In ters of SI units ( ) ( ) ( ) And in ters of diensions kg s kg s ( ) ( ) ( ) a b c M T M T M T For each diension (M, or T) the owers ust be equal on both sides of the equation, so for M: 0 a + a - for : 0 -a + b + c b + c for T: 0 -b - b - c -4 - b - Giving π as π u d F π u d F CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 6

6 And a siilar rocedure is followed for the other π grous. Grou π a u b d c N ( ) ( ) ( ) a b c M T M T T For each diension (M, or T) the owers ust be equal on both sides of the equation, so for M: 0 a for : 0 -a + b + c 0 b + c for T: 0 -b - b - c Giving π as π π 0 u d N Nd u And for the third, π a u b d c ( ) ( ) ( ) a b c M T M T M T For each diension (M, or T) the owers ust be equal on both sides of the equation, so for M: 0 a + a - for : 0 -a + b + c - b + c - for T: 0 -b - b - c - Giving π as π u d π ud Thus the roble ay be described by the following function of the three non-diensional π grous, φ ( π, π, π ) 0 CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 7

7 F Nd φ,, u d u ud 0 This ay also be written: F Nd u d φ, u ud 5.6. Wrong choice of hysical roerties. If, when defining the roble, extra - uniortant - variables are introduced then extra π grous will be fored. They will lay very little role influencing the hysical behaviour of the roble concerned and should be identified during exeriental work. If an iortant / influential variable was issed then a π grou would be issing. Exeriental analysis based on these results ay iss significant behavioural changes. It is therefore, very iortant that the initial choice of variables is carried out with great care. 5.7 Maniulation of the π grous Once identified aniulation of the π grous is eritted. These aniulations do not change the nuber of grous involved, but ay change their aearance drastically. Taking the defining equation as: φ ( π, π, π π -n ) 0 Then the following aniulations are eritted: i. Any nuber of grous can be cobined by ultilication or division to for a new grou which relaces one of the existing. E.g. π and π ay be cobined to for π a π / π so the defining equation becoes φ ( π a, π, π π -n ) 0 ii. The recirocal of any diensionless grou is valid. So φ ( π,/ π, π /π -n ) 0 is valid. iii. Any diensionless grou ay be raised to any ower. So φ ( (π ), (π ) /, (π ) π -n ) 0 is valid. iv. Any diensionless grou ay be ultilied by a constant. v. Any grou ay be exressed as a function of the other grous, e.g. π φ ( π, π π -n ) In general the defining equation could look like φ ( π, /π,( π ) i 0.5π -n ) Coon π grous During diensional analysis several grous will aear again and again for different robles. These often have naes. You will recognise the eynolds nuber ud/. Soe coon non-diensional nubers (grous) are listed below. eynolds nuber e ud inertial, viscous force ratio CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 8

8 Euler nuber En u ressure, inertial force ratio Froude nuber Weber nuber Fn u gd We ud σ inertial, gravitational force ratio inertial, surface tension force ratio Mach nuber Mn u c ocal velocity, local velocity of sound ratio 5.9 Exales The discharge Q through an orifice is a function of the diaeter d, the ressure difference, the density, and the viscosity, show that Q d / d / / / φ, where φ is soe unknown function. Write out the diensions of the variables : M - u: T - d: : M - T - :(force/area) M - T - We are told fro the question that there are 5 variables involved in the roble: d,,, and Q. Choose the three recurring (governing) variables; Q, d,. Fro Buckingha s π theore we have -n 5 - non-diensional grous. ( Q, d,,, ) φ( π, π ) φ a b c π Q d π For the first grou, π : 0 0 Q d a b c ( ) ( ) ( ) a b c M T T M M T M] 0 c + c - ] 0 a + b - c - - a + b T] 0 -a - a - b π Q d d Q CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 9

9 And the second grou π : (note is a ressure (force/area) with diensions M - T - ) ( ) ( ) ( ) a b c M T T M MT M] 0 c + c - ] 0 a + b - c - - a + b T] 0 -a - a - b 4 4 π Q d 4 d Q So the hysical situation is described by this function of non-diensional nubers, 4 φ( π, π ) φ, 0 d d Q Q or 4 d φ d Q Q The question wants us to show : Q d / d / φ / / Take the recirocal of square root of π : π / Q π d, / a Convert π by ultilying by this new grou, π a π / d Q π π a Q d d a / / / then we can say / / / d d φ( / π a, πa ) φ, / 0 Q or / Q d / φ d / / CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 0

10 5.0 Siilarity Hydraulic odels ay be either true or distorted odels. True odels reroduce features of the rototye but at a scale - that is they are geoetrically siilar Geoetric siilarity Geoetric siilarity exists between odel and rototye if the ratio of all corresonding diensions in the odel and rototye are equal. odel rototye λ where λ is the scale factor for length. For area A A odel rototye λ All corresonding angles are the sae Kineatic siilarity Kineatic siilarity is the siilarity of tie as well as geoetry. It exists between odel and rototye i. If the aths of oving articles are geoetrically siilar ii. If the rations of the velocities of articles are siilar Soe useful ratios are: Velocity V V / T u T λ / λ λ T Acceleration a a / T λ λa / T λ T Discharge Q Q / T λ λq / T λ T This has the consequence that strealine atterns are the sae Dynaic siilarity Dynaic siilarity exists between geoetrically and kineatically siilar systes if the ratios of all forces in the odel and rototye are the sae. CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity

11 Force ratio F F M a λ λ λ λ λλλu M a λ λ T T This occurs when the controlling diensionless grou on the right hand side of the defining equation is the sae for odel and rototye. 5. Models When a hydraulic structure is build it undergoes soe analysis in the design stage. Often the structures are too colex for sile atheatical analysis and a hydraulic odel is build. Usually the odel is less than full size but it ay be greater. The real structure is known as the rototye. The odel is usually built to an exact geoetric scale of the rototye but in soe cases - notably river odel - this is not ossible. Measureents can be taken fro the odel and a suitable scaling law alied to redict the values in the rototye. To illustrate how these scaling laws can be obtained we will use the relationshi for resistance of a body oving through a fluid. The resistance,, is deendent on the following hysical roerties: : M - u: T - l:(length) : M - T - So the defining equation is φ (,, u, l, ) 0 Thus, 5, n so there are 5- π grous π a u b l c π a u b d c For the π grou ( ) ( ) ( ) eading to π as π u l a b c M T M T M T a b c For the π grou ( ) ( ) ( ) eading to π as π ul M T M T M T Notice how /π is the eynolds nuber. We can call this π a. So the defining equation for resistance to otion is We can write φ ( π, π a ) 0 CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity

12 u l ul φ ul u l φ This equation alies whatever the size of the body i.e. it is alicable to a to the rototye and a geoetrically siilar odel. Thus for the odel ul φ u l and for the rototye ul φ u l Dividing these two equations gives / u l / u l ( u l / ) φ ( ul / ) φ At this oint we can go no further unless we ake soe assutions. One coon assution is to assue that the eynolds nuber is the sae for both the odel and rototye i.e. u l / u l / This assution then allows the equation following to be written ul u l Which gives this scaling law for resistance force: λ λ λ λ u That the eynolds nubers were the sae was an essential assution for this analysis. The consequence of this should be exlained. e u l e u l u u λ λu λ λ l l Substituting this into the scaling law for resistance gives CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity

13 λ λ λ λ So the force on the rototye can be redicted fro easureent of the force on the odel. But only if the fluid in the odel is oving with sae eynolds nuber as it would in the rototye. That is to say the can be redicted by u l u l rovided that u l l u In this case the odel and rototye are dynaically siilar. Forally this occurs when the controlling diensionless grou on the right hand side of the defining equation is the sae for odel and rototye. In this case the controlling diensionless grou is the eynolds nuber. 5.. Dynaically siilar odel exales Exale An underwater issile, diaeter and length 0 is tested in a water tunnel to deterine the forces acting on the real rototye. A /0 th scale odel is to be used. If the axiu allowable seed of the rototye issile is 0 /s, what should be the seed of the water in the tunnel to achieve dynaic siilarity? For dynaic siilarity the eynolds nuber of the odel and rototye ust be equal: e e ud ud So the odel velocity should be u u d d As both the odel and rototye are in water then, and so u u d s d / / Note that this is a very high velocity. This is one reason why odel tests are not always done at exactly equal eynolds nubers. Soe relaxation of the equivalence requireent is often accetable when the eynolds nuber is high. Using a wind tunnel ay have been ossible in this exale. If this were the case then the aroriate values of the and ratios need to be used in the above equation. Exale CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 4

14 A odel aerolane is built at /0 scale and is to be tested in a wind tunnel oerating at a ressure of 0 ties atosheric. The aerolane will fly at 500k/h. At what seed should the wind tunnel oerate to give dynaic siilarity between the odel and rototye? If the drag easure on the odel is 7.5 N what will be the drag on the lane? Fro earlier we derived the equation for resistance on a body oving through air: ul u l φ u l φ( e) For dynaic siilarity e e, so u u d d The value of does not change uch with ressure so The equation of state for an ideal gas is T. As teerature is the sae then the density of the air in the odel can be obtained fro 0 T T 0 So the odel velocity is found to be u u 0. 5u 0 / 0 u 50k / h The ratio of forces is found fro ( u l ) ( u l ) ( 05. ) ( 0. ) So the drag force on the rototye will be N CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 5

15 5.. Models with free surfaces - rivers, estuaries etc. When odelling rivers and other fluid with free surfaces the effect of gravity becoes iortant and the ajor governing non-diensional nuber becoes the Froude (Fn) nuber. The resistance to otion forula above would then be derived with g as an extra deendent variables to give an extra π grou. So the defining equation is: φ (,, u, l,, g ) 0 Fro which diensional analysis gives: ul u u l φ, gl u l φ ( e,fn) Generally the rototye will have a very large eynolds nuber, in which case slight variation in e causes little effect on the behaviour of the roble. Unfortunately odels are soeties so sall and the eynolds nubers are large and the viscous effects take effect. This situation should be avoided to achieve correct results. Solutions to this roble would be to increase the size of the odel - or ore difficult - to change the fluid (i.e. change the viscosity of the fluid) to reduce the e. 5.. Geoetric distortion in river odels When river and estuary odels are to be built, considerable robles ust be addressed. It is very difficult to choose a suitable scale for the odel and to kee geoetric siilarity. A odel which has a suitable deth of flow will often be far to big - take u too uch floor sace. educing the size and retaining geoetric siilarity can give tiny deth where viscous force coe into lay. These result in the following robles: i. accurate deths and deth changes becoe very difficult to easure; ii. the bed roughness of the channel becoes iracticably sall; iii. lainar flow ay result - (turbulent flow is noral in river hydraulics.) The solution often adoted to overcoe these robles is to abandon strict geoetric siilarity by having different scales in the horizontal and the vertical. Tyical scales are /00 in the vertical and between /00 and /500 in the horizontal. Good overall flow atterns and discharge characteristics can be roduced by this technique, however local detail of flow is not well odelled. In these odel the Froude nuber (u /d) is used as the doinant non-diensional nuber. Equivalence in Froude nubers can be achieved between odel and rototye even for distorted odels. To address the roughness roble artificially high surface roughness of wire esh or sall blocks is usually used. CIVE 400: Fluid Mechanics Diensional Analysis and Siilarity 6