# HW11 Solution

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Homework #11 ME 363 - Fluid Mechanics Due Wednesday, April 16, 2008Fall Semester 20081]A student team is to desin a !uman powered sumarine \$or a desin %ompetition& '!e o(erall lent! o\$ t!e  prototype sumarine is 2&2) m, and its student desiners !ope t!at it %an tra(el \$ully sumered t!rou! water at0&*6 m+s& '!e water is \$res!water a lake- at ' . 1* derees /& '!e desin team uilds a one ei!t! s%ale model to test in t!eir uni(ersitys wind tunnel& A s!ield surrounds t!e dra alan%e strut so t!at t!e aerodynami%dra o\$ t!e strut itsel\$ does not in\$luen%e t!e measured dra& '!e air in t!e wind tunnel is at 2* derees / and atatmosp!eri% pressure& At w!at air speed do t!ey need to run t!e wind tunnel in order to a%!ie(e similarity2]'!is prolem uilds on prolem 1& '!e students measure t!e aerodynami% dra on t!eir model sumarine in t!ewind tunnel& '!ey are %are\$ul to run t!e wind tunnel at %onditions t!at ensure similarity wit! t!e prototype sumarine& '!eir measured dra \$or%e is 2& 3& 4stimate t!e dra \$or%e on t!e prototype sumarine at t!e %onditions i(en in prolem 1& ]Some students want to (isuali5e \$low o(er a spinnin aseall& '!eir \$luids laoratory !as a ni%e water tunnel into w!i%! t!ey %an ine%t multi%olored dye streaklines, so t!ey de%ide to test a spinnin aseall in t!e water tunnel& Similarity re7uires t!at t!ey mat%! ot! t!e eynolds numer and t!e Strou!al numer etween t!eir model test and t!e a%tual aseall t!at mo(es t!rou! t!e air at 80 mp! and spins at 00 rpm& 9ot! t!e air and t!e water are at 20 derees /& At w!at speed s!ould t!ey run t!e water in t!e water tunnel, and at w!at rpm s!ould t!ey spin t!eir aseallno \$iure:rolem #1:rolem #2:rolem #  :rolem 1 Solution For a s%ale model o\$ a sumarine ein tested in air, we are to %al%ulate t!e wind tunnel speedre7uired to a%!ie(e similarity wit! t!e prototype sumarine t!at mo(es t!rou! water at a i(en speed&  Assumptions 1  /ompressiility o\$ t!e air is assumed to e neliile& 2  '!e wind tunnel walls are \$ar enou!away so as to not inter\$ere wit! t!e aerodynami% dra on t!e model su& 3  '!e model is eometri%ally similar to t!e prototype&  Properties For water at T   . 1* o / and atmosp!eri% pressure,  ρ   . ;;;&1 k+m   and  µ   . 1&18 ×  10   k+m ⋅ s& For air at T   . 2* o / and atmosp!eri% pressure,  ρ   . 1&18) k+m   and  µ   . 1&8); ×  10 *  k+m ⋅ s&  Analysis Similarity is a%!ie(ed w!en t!e eynolds numer o\$ t!e model is e7ual to t!at o\$ t!e prototype, Similarity <  pppmmmmpmp 8e 8e V LV L  ρ  ρ  µ µ  = = = 1-We sol(e 47& 1 \$or t!e unknown wind tunnel speed, ( ) ( )  ppmmp pmm* 1&8);10 k+ms;;;&1 k+m 0&*60 m+s81&1810 k+ms1&18) k+m  LV V  L  ρ  µ  µ ρ  −−       =  ÷  ÷ ÷ ÷        × ×= = ÷ ÷× ×    61.4 m/s  Discussion At t!is air temperature, t!e speed o\$ sound is around )6 m+s& '!us t!e =a%! numer in t!e windtunnel is e7ual to 61&)+)6 . 0&1>>& '!is is su\$\$i%iently low t!at t!e in%ompressile \$low appro?imation isreasonale&:rolem 2 Solution We are to estimate t!e dra on a prototype sumarine in water, ased on aerodynami% drameasurements per\$ormed in a wind tunnel&  Assumptions 1  '!e model is eometri%ally similar& 2  '!e wind tunnel is run at %onditions w!i%! ensure similarity etween model and prototype&  Properties For water at T   . 1* o / and atmosp!eri% pressure,  ρ   . ;;;&1 k+m   and  µ   . 1&18 ×  10   k+m ⋅ s& For air at T   . 2* o / and atmosp!eri% pressure,  ρ   . 1&18) k+m   and  µ   . 1&8); ×  10 *  k+m ⋅ s&  Analysis Sin%e t!e eynolds numers !a(e een mat%!ed, t!e nondimensionali5ed dra %oe\$\$i%ient o\$ t!emodel e7uals t!at o\$ t!e prototype, ,p,m2222mmmppp  D D  F  F V L V L  ρ ρ  = 1-We sol(e 47& 1 \$or t!e unknown aerodynami% dra \$or%e on t!e prototype,  F   D,  p , ( ) ( ) 2222 ppp,p,mmmm ;;;&1 k+ms0&*60 m+s2& 381&18) k+ms61&) m+s  D D V L F F V L  ρ  ρ          ×   = = = ÷ ÷  ÷  ÷  ÷×           10.3 N w!ere we !a(e used t!e wind tunnel speed %al%ulated in :rolem > 6&  Discussion Alt!ou! t!e prototype mo(es at a mu%! slower speed t!an t!e model, t!e density o\$ water is mu%!!i!er t!an t!at o\$ air, and t!e prototype is ei!t times larer t!an t!e model& W!en all o\$ t!ese \$a%tors are%omined, t!e dra \$or%e on t!e prototype is mu%! larer t!an t!at on t!e model&  :rolem  Solution We are to %al%ulate t!e speed and anular (elo%ity rpm- o\$ a spinnin aseall in a water %!annelsu%! t!at \$low %onditions are dynami%ally similar to t!at o\$ t!e a%tual aseall mo(in and spinnin in air&  Properties For air at T   . 20 o / and atmosp!eri% pressure,  ρ   . 1&20) k+m   and  µ   . 1&82* ×  10 *  k+m ⋅ s& For water at T   . 20 o / and atmosp!eri% pressure,  ρ   . ;;8&0 k+m   and  µ   . 1&002 ×  10   k+m ⋅ s&  Analysis '!e model in t!e water- and t!e prototype in t!e air- are a%tually t!e same aseall, so t!eir %!ara%teristi% lent!s are e7ual,  L m  .  L  p & We mat%! eynolds numer,  pppmmmmpmp 8e 8e V LV L  ρ  ρ  µ µ  = = = 1-and sol(e \$or t!e re7uired water tunnel speed \$or t!e model tests, V  m , ( ) ( )  ppmmp* pmm 1&00210 k+ms1&20) k+m80&0 mp!11&82*10 k+ms;;8&0 k+m  LV V  L  ρ  µ  µ ρ  −−           × ×= = = ÷  ÷ ÷  ÷ ÷ ÷ × ×         5.30 mph 2-We also mat%! Strou!al numers, re%oni5in t!at n & is proportional to  f   ,  ppmmmpmp St St  f L f LV V  = = =   →    ppmmmp n Ln LV V  = && -\$rom w!i%! we sol(e \$or t!e re7uired spin rate in t!e water tunnel, ( ) ( )  pmmpmp *&0 mp!00 rpm180&0 mp!  L V n n L V         = = = ÷ ÷  ÷ ÷        19.9 rpm & & )-  Discussion 9e%ause o\$ t!e di\$\$eren%e in \$luid properties etween air and water, t!e re7uired water tunnel speedis mu%! lower t!an t!at in air& @n addition, t!e spin rate is mu%! lower, makin \$low (isuali5ation easier&

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