TOPIC T3: DIMENSIONAL ANALYSIS AUTUMN 013 Objectives (1 Be able to deterine the diensions of hysical quantities in ters of fundaental diensions. ( Understand the Princile of Diensional Hoogeneity and its use in checking equations and reducing hysical robles. (3 Be able to carry out a foral diensional analysis using Buckingha s Pi Theore. (4 Understand the requireents of hysical odelling and its liitations. 1. What is diensional analysis?. Diensions.1 Diensions and units. Priary diensions.3 Diensions of derived quantities.4 Working out diensions.5 Alternative choices for riary diensions 3. Foral rocedure for diensional analysis 3.1 Diensional hoogeneity 3. Buckingha s Pi theore 3.3 Alications 4. Physical odelling 4.1 Method 4. Incolete siilarity ( scale effects 4.3 Froude-nuber scaling 5. Non-diensional grous in fluid echanics References White (011 Chater 5 Haill (011 Chater 10 Chadwick and Morfett (013 Chater 11 Massey (011 Chater 5 Hydraulics T3-1 David Asley
1. WHAT IS DIMENSIONAL ANALYSIS? Diensional analysis is a eans of silifying a hysical roble by aealing to diensional hoogeneity to reduce the nuber of relevant variables. It is articularly useful for: resenting and interreting exeriental data; attacking robles not aenable to a direct theoretical solution; checking equations; establishing the relative iortance of articular hysical henoena; hysical odelling. Exale. The drag force F er unit length on a long sooth cylinder is a function of air seed U, density ρ, diaeter D and viscosity μ. However, instead of having to draw hundreds of grahs ortraying its variation with all cobinations of these araeters, diensional analysis tells us that the roble can be reduced to a single diensionless relationshi c D f (Re where c D is the drag coefficient and Re is the Reynolds nuber. In this instance diensional analysis has reduced the nuber of relevant variables fro 5 to and the exeriental data to a single grah of c D against Re.. DIMENSIONS.1 Diensions and Units A diension is the tye of hysical quantity. A unit is a eans of assigning a nuerical value to that quantity. SI units are referred in scientific work.. Priary Diensions In fluid echanics the riary or fundaental diensions, together with their SI units are: ass M (kilogra, kg length L (etre, tie T (second, s teerature Θ (kelvin, K In other areas of hysics additional diensions ay be necessary. The colete set secified by the SI syste consists of the above lus electric current I (aere, A luinous intensity C (candela, cd aount of substance n (ole, ol Hydraulics T3- David Asley
.3 Diensions of Derived Quantities Diensions of coon derived echanical quantities are given in the following table. Geoetry Kineatics Dynaics Fluid roerties Quantity Coon Sybol(s Diensions Area A L Volue V L 3 Second oent of area I L 4 Velocity U LT 1 Acceleration a LT Angle θ 1 (i.e. diensionless Angular velocity ω T 1 Quantity of flow Q L 3 T 1 Mass flow rate MT 1 Force F MLT Moent, torque T ML T Energy, work, heat E, W ML T Power P ML T 3 Pressure, stress, τ ML 1 T Density ρ ML 3 Viscosity μ ML 1 T 1 Kineatic viscosity ν L T 1 Surface tension σ MT Theral conductivity k MLT 3 Θ 1 Secific heat c, c v L T Θ 1 Bulk odulus K ML 1 T.4 Working Out Diensions In the following, [ ] eans diensions of. Exale. Use the definition du τ μ to deterine the diensions of viscosity. dy Solution. Fro the definition, τ force / area μ du /dy velocity / length Hence, MLT /L 1 1 [μ] ML T 1 LT /L Alternatively, diensions ay be deduced indirectly fro any known forula involving that quantity. Hydraulics T3-3 David Asley
Exale. ρul Since Re is known to be diensionless, the diensions of μ ust be the sae as μ those of ρul; i.e. 3 1 1 1 [μ] [ρul] (ML (LT (L ML T.5 Alternative Choices For Priary Diensions The choice of riary diensions is not unique. It is not uncoon and it ay soeties be ore convenient to choose force F as a riary diension rather than ass, and have a {FLT} syste rather than {MLT}. Exale. Find the diensions of viscosity μ in the {FLT} rather than {MLT} systes. Answer: [μ] = FL T Hydraulics T3-4 David Asley
3. FORMAL PROCEDURE FOR DIMENSIONAL ANALYSIS 3.1 Diensional Hoogeneity The Princile of Diensional Hoogeneity All additive ters in a hysical equation ust have the sae diensions. Exales: 1 s ut at all ters have the diensions of length (L V ρ g g z H all ters have the diensions of length (L Diensional hoogeneity is a useful tool for checking forulae. For this reason it is useful when analysing a hysical roble to retain algebraic sybols for as long as ossible, only substituting nubers right at the end. However, diensional analysis cannot deterine nuerical factors; e.g. it cannot distinguish between ½at and at in the first forula above. Diensional hoogeneity is the basis of the foral diensional analysis that follows. 3. Buckingha s Pi Theore Exerienced ractitioners can do diensional analysis by insection. However, the foral tool which they are unconsciously using is Buckingha s Pi Theore 1 : Buckingha s Pi Theore (1 If a roble involves n relevant variables indeendent diensions then it can be reduced to a relationshi between n non-diensional araeters Π 1,..., Π n-. ( To construct these non-diensional Π grous: (i Choose diensionally-distinct scaling variables (aka reeating variables. (ii For each of the n reaining variables construct a non-diensional Π of the for a b c Π ( variable ( scale 1 ( scale ( scale 3 where a, b, c,... are chosen so as to ake each Π non-diensional. Note. In order to ensure diensional indeendence in {MLT} systes it is coon but not obligatory to choose the scaling variables as: a urely geoetric quantity (e.g. a length, a kineatic (tie- but not ass-containing quantity (e.g. velocity or acceleration and a dynaic (ass- or force-containing quantity (e.g. density. 1 Buckingha, E., 1914. The use of Π coes fro its use as the atheatical sybol for a roduct. Hydraulics T3-5 David Asley
3.3 Alications Exale. Obtain an exression in non-diensional for for the ressure gradient in a horizontal ie of circular cross-section. Show how this relates to the failiar exression for frictional head loss. Ste 1. Identify the relevant variables. d/dx, ρ, V, D, k s, μ Ste. Write down diensions. force / area] dx length ρ ML 3 V LT 1 D L k s L μ ML 1 T 1 d [ MLT L L ML Ste 3. Establish the nuber of indeendent diensions and non-diensional grous. Nuber of relevant variables: n = 6 Nuber of indeendent diensions: = 3 (M, L and T Nuber of non-diensional grous (Πs: n = 3 Ste 4. Choose (= 3 diensionally-indeendent scaling variables. e.g. geoetric (D, kineatic/tie-deendent (V, dynaic/ass-deendent (ρ. Ste 5. Create the Πs by non-diensionalising the reaining variables: d/dx, k s and μ. d a c Π D V b 1 ρ dx Considering the diensions of both sides: 0 0 0 a 1 b 3 c M L T (ML T (L (LT (ML 1c ab3c b M L T Equate owers of riary diensions. Since M only aears in [ρ] and T only aears in [V] it is sensible to deal with these first. M: 0 = 1 + c c = 1 T: 0 = b b = L: 0 = + a + b 3c a = b + 3c = 1 Hence, d D d 1 Π d 1 DV ρ x (Check: OK ratio of two ressures dx ρv T k s Π (by insection, since k s is a length D Hydraulics T3-6 David Asley
a c Π μd V b 3 ρ In ters of diensions: 0 0 0 1 M L T (ML T 1 (L a (LT 1 b (ML 1c 1 ab3c 1b M L T Equating exonents: M: 0 = 1 + c c = 1 T: 0 = 1 b b = 1 L: 0 = 1 + a + b 3c a = 1 b + 3c = 1 Hence, μ Π 3 ρvd (Check: OK this is the recirocal of the Reynolds nuber 3 c Ste 6. Set out the non-diensional relationshi. Π1 f (Π,Π3 or d D dx ks μ f (, (* ρv D ρvd Ste 7. Rearrange (if required for convenience. We are free to relace any of the Πs by a ower of that Π, or by a roduct with the other Πs, rovided we retain the sae nuber of indeendent diensionless grous. In this case we recognise that Π 3 is the recirocal of the Reynolds nuber, so it looks 1 better to use Π (Π Re as the third non-diensional grou. We can also write 3 3 the ressure gradient in ters of head loss: d h f ρg. With these two odifications dx L the non-diensional relationshi (* then becoes gh f D ks f (,Re LV D or L V ks h f f (,Re D g D Since nuerical factors can be absorbed into the non-secified function, this can easily be identified with the Darcy-Weisbach equation h L V λ f D g where λ is a function of relative roughness k s /D and Reynolds nuber Re, a function given (Toic by the Colebrook-White equation. Hydraulics T3-7 David Asley
Exale. The drag force on a body in a fluid flow is a function of the body size (exressed via a characteristic length L and the fluid velocity V, density ρ and viscosity μ. Perfor a diensional analysis to reduce this to a single functional deendence c D f (Re where c D is a drag coefficient and Re is the Reynolds nuber. What additional non-diensional grous ight aear in ractice? Notes. (1 Diensional analysis sily says that there is a relationshi; it doesn t (excet in the case of a single Π, which ust, therefore, be constant say what the relationshi is. For the secific relationshi one ust aeal to theory or, ore coonly, exeriental data. ( If Π 1, Π, Π 3,... are suitable non-diensional grous then we are liberty to relace soe or all of the by any owers or roducts with the other Πs, rovided that we retain the sae nuber of indeendent non-diensional grous; e.g. (Π 1 1, (Π, Π 1 /(Π 3. (3 It is extreely coon in fluid echanics to find (often after the rearrangeent entioned in ( certain cobinations which can be recognised as key araeters such as the Reynolds nuber ( Re ρul/ μ or Froude nuber ( Fr U/ gl. (4 Often the hardest art of the diensional analysis is deterining which are the relevant variables. For exale, surface tension is always resent in free-surface flows, but can be neglected if the Weber nuber We = ρu L/σ is large. Siilarly, all fluids are coressible, but coressibility effects on the flow can be ignored if the Mach nuber (Ma = U/c is sall; i.e. velocity is uch less than the seed of sound. (5 Although three riary diensions (M,L,T ay aear when the variables are listed, they do not do so indeendently. The following exale illustrates a case where M and T always aear in the cobination MT, hence giving only one indeendent diension. Hydraulics T3-8 David Asley
Exale. The ti deflection δ of a cantilever bea is a function of ti load W, bea length l, second oent of area I and Young s odulus E. Perfor a diensional analysis of this roble. Ste 1. Identify the relevant variables. δ, W, l, I, E. Ste. Write down diensions. δ L W MLT l L I L 4 E ML 1 T Ste 3. Establish the nuber of indeendent diensions and non-diensional grous. Nuber of relevant variables: n = 5 Nuber of indeendent diensions: = (L and MT - note Nuber of non-diensional grous (Πs: n = 3 Ste 4. Choose (= diensionally-indeendent scaling variables. e.g. geoetric (l, ass- or tie-deendent (E. Ste 5. Create the Πs by non-diensionalising the reaining variables: δ, I and W. These give (after soe algebra, not reroduced here: δ Π 1 l I Π 4 l W Π 3 El Ste 6. Set out the non-diensional relationshi. Π1 f (Π,Π3 or δ ( I, W f 4 l l El This is as far as diensional analysis will get us. Detailed theory shows that, for sall elastic deflections, 1 Wl 3 δ 3 EI or δ 1 W l 3 El I 4 l 1 Hydraulics T3-9 David Asley
4. PHYSICAL MODELLING 4.1 Method If a diensional analysis indicates that a roble is described by a functional relationshi between non-diensional araeters Π 1, Π, Π 3,... then full siilarity requires that these araeters be the sae at both full ( rototye scale and odel scale. i.e. Π (Π ( 1 1 ( Π (Π etc. Exale. A rototye gate valve which will control the flow in a ie syste conveying araffin is to be studied in a odel. List the significant variables on which the ressure dro across the valve would deend. Perfor diensional analysis to obtain the relevant non-diensional grous. A 1/5 scale odel is built to deterine the ressure dro across the valve with water as the working fluid. (a For a articular oening, when the velocity of araffin in the rototye is 3.0 s 1 what should be the velocity of water in the odel for dynaic siilarity? (b (c What is the ratio of the quantities of flow in rototye and odel? Find the ressure dro in the rototye if it is 60 kpa in the odel. (The density and viscosity of araffin are 800 kg 3 and 0.00 kg 1 s 1 resectively. Take the kineatic viscosity of water as 1.0 10 6 s 1. Solution. The ressure dro Δ is exected to deend uon the gate oening h, the overall deth d, the velocity V, density ρ and viscosity μ. List the relevant variables: Δ, h, d, V, ρ, μ Write down diensions: Δ ML 1 T h L d L V LT 1 ρ ML 3 μ ML 1 T 1 Nuber of variables: n = 6 Nuber of indeendent diensions: = 3 (M, L and T Nuber of non-diensional grous: n = 3 Hydraulics T3-10 David Asley
Choose (= 3 scaling variables: geoetric (d; kineatic/tie-deendent (V; dynaic/ass-deendent (ρ. For diensionless grous by non-diensionalising the reaining variables: Δ, h and μ. Π 1 0 0 0 M L T a c Δd V b ρ 1 (ML T (L a (LT 1 b (ML 1c 1 ab3c b M L T M: 0 = 1 + c c = 1 T: 0 = b b = L: 0 = 1 + a + b 3c a = 1 + 3c b = 0 1 Δ Π1 ΔV ρ ρv 3 c h Π (by insection, since h is a length d a c Π μd V b 3 ρ (robably obvious by now, but here goes anyway... 0 0 0 M L T 1 (ML T 1 (L a (LT 1 b (ML 1c 1 ab3c 1b M L T M: 0 = 1 + c c = 1 T: 0 = 1 b + 0 b = 1 L: 0 = 1 +a + b 3c a = 1 + 3c b = 1 1 1 1 μ Π3 μd V ρ ρvd Recognition of the Reynolds nuber suggests that we relace Π 3 by 1 ρvd Π3 (Π3 μ 3 c Hence, diensional analysis yields Π1 f (Π,Π3 i.e. Δ h ρvd (, f ρv d μ (a Dynaic siilarity requires that all non-diensional grous be the sae in odel and rototye; i.e. Δ Δ Π V V 1 ρ ρ h h Π (autoatic if siilar shae; i.e. geoetric siilarity d d Hydraulics T3-11 David Asley
ρvd ρvd Π 3 μ μ Fro the last, we have a velocity ratio V (μ/ρ d 0.00 /800 1 V (μ/ρ d 1.0 10 5 Hence, V 3.0 1 V 6.0 s 0.5 0.5 6 0.5 (b The ratio of the quantities of flow is Q Q ( V d velocity area ( velocity area V d 0.5 5 1.5 (c Finally, for the ressure dro, Δ Δ (Δ ρ V 800 Π 0.5 0. ρ ρ (Δ ρ V V V 1000 Hence, Δ 0. Δ 0. 60 1.0 kpa Hydraulics T3-1 David Asley
4. Incolete Siilarity ( Scale Effects For a ulti-araeter roble it is often not ossible to achieve full siilarity. In articular, it is rare to be able to achieve full Reynolds-nuber scaling when other diensionless araeters are also involved. For hydraulic odelling of flows with a free surface the ost iortant requireent is Froude-nuber scaling (Section 4.3 It is coon to distinguish three levels of siilarity. Geoetric siilarity the ratio of all corresonding lengths in odel and rototye are the sae (i.e. they have the sae shae. Kineatic siilarity the ratio of all corresonding lengths and ties (and hence the ratios of all corresonding velocities in odel and rototye are the sae. Dynaic siilarity the ratio of all forces in odel and rototye are the sae; e.g. Re = (inertial force / (viscous force is the sae in both. Geoetric siilarity is alost always assued. However, in soe alications notably river odelling it is necessary to distort vertical scales to revent undue influence of, for exale, surface tension or bed roughness. Achieving full siilarity is articularly a roble with the Reynolds nuber Re = UL/ν. Using the sae working fluid would require a velocity ratio inversely roortional to the length-scale ratio and hence iractically large velocities in the scale odel. A velocity scale fixed by, for exale, the Froude nuber (see Section 4.3 eans that the only way to aintain the sae Reynolds nuber is to adjust the kineatic viscosity (substantially. In ractice, Reynolds-nuber siilarity is uniortant if flows in both odel and rototye are fully turbulent; then oentu transort by viscous stresses is uch less than that by turbulent eddies and so the recise value of olecular viscosity μ is uniortant. In soe cases this ay ean deliberately triggering transition to turbulence in boundary layers (for exale by the use of triing wires or roughness stris. Surface effects Full geoetric siilarity requires that not only the ain diensions of objects but also the surface roughness and, for obile beds, the sedient size be in roortion. This would ut iossible requireents on surface finish or grain size. In ractice, it is sufficient that the surface be aerodynaically rough: u /ν τks 5, where uτ τ w / ρ is the friction velocity and k s a tyical height of surface irregularities. This ioses a iniu velocity in odel tests. Other Fluid Phenoena When scaled down in size, fluid henoena which were negligible at full scale ay becoe iortant in laboratory odels. A coon exale is surface tension. Hydraulics T3-13 David Asley
4.3 Froude-Nuber Scaling The ost iortant araeter to reserve in hydraulic odelling of free-surface flows driven by gravity is the Froude nuber, Fr U / gl. Preserving this araeter between odel ( and rototye ( dictates the scaling of other variables in ters of the length scale ratio. Velocity ( Fr (Fr U U U L gl gl U L i.e. the velocity ratio is the square root of the length-scale ratio. Quantity of flow Force Q ~ velocity area Q Q L L F F ~ ressure area L F L This arises since the ressure ratio is equal to the length-scale ratio this can be seen fro either hydrostatics (ressure height or fro the dynaic ressure (roortional to (velocity which, fro the Froude nuber, is roortional to length. 5 / 1/ 3 Tie t ~ length velocity t t L L 1/ Hence the quantity of flow scales as the length-scale ratio to the 5/ ower, whilst the tie-scale ratio is the square root of the length-scale ratio. For exale, at 1:100 geoetric scale, a full-scale tidal eriod of 1.4 hours becoes 1.4 hours. Exale. The force exerted on a bridge ier in a river is to be tested in a 1:10 scale odel using water as the working fluid. In the rototye the deth of water is.0, the velocity of flow is 1.5 s 1 and the width of the river is 0. (a (b (c List the variables affecting the force on the ier and erfor diensional analysis. Can you satisfy all the conditions for colete siilarity? What is the ost iortant araeter to choose for dynaic siilarity? What are the deth, velocity and quantity of flow in the odel? If the hydrodynaic force on the odel bridge ier is 5 N, what would it be on the rototye? Hydraulics T3-14 David Asley
5. NON-DIMENSIONAL GROUPS IN FLUID MECHANICS Dynaic siilarity requires that the ratio of all forces be the sae. The ratio of different forces roduces any of the key non-diensional araeters in fluid echanics. (Note that inertial force eans ass acceleration since it is equal to the total alied force it is often one of the two forces in the ratio. Reynolds nuber Froude nuber Weber nuber Rossby nuber Mach nuber ρul Re μ Fr U gl ρu L We σ U Ro ΩL U Ma c inertial force (viscous flows viscous force inertial force gravitational force 1/ (free-surface flows inertial force surface tension (surface tension inertial force Coriolis force (rotating flows inertial force coressibility forc e 1/ (coressible flows These grous occur regularly when diensional analysis is alied to fluid-dynaical robles. They can be derived by considering forces on a sall volue of fluid. They can also be derived by non-diensionalising the differential equations of fluid flow (see White, Chater 5, or the online notes for the 4 th -year Coutational Hydraulics unit. Hydraulics T3-15 David Asley