Ис­то­рия гид­ро­ди­на­ми­ки

Не­по­знан­ная тур­бу­лент­ность

Од­на­ко вско­ре вы­яс­ни­лось, что в по­дав­ляю­щем боль­шин­стве ре­аль­ных слу­ча­ев ла­ми­нар­ное те­че­ние сла­бо­вяз­кой жид­ко­сти ста­но­вит­ся не­устой­чи­вым и ра­но или позд­но пе­ре­хо­дит в тур­бу­лент­ное [3]:

Турбуле́нтность [...] — яв­ле­ние, за­клю­чаю­щее­ся в том, что при пре­вы­ше­нии не­ко­то­ро­го кри­ти­че­ско­го чис­ла Рей­нольд­са и/или Ре­лея (в част­ном слу­чае при пре­вы­ше­нии ско­ро­сти по­то­ка при по­сто­ян­ной плот­но­сти и диа­мет­ре тру­бы и/или тем­пе­ра­ту­ры на внеш­ней гра­ни­це сре­ды) са­мо­про­из­воль­но об­ра­зу­ют­ся мно­го­чис­лен­ные не­ли­ней­ные фрак­таль­ные вол­ны и обыч­ные, ли­ней­ные раз­лич­ных раз­ме­ров, без на­ли­чия внеш­них, слу­чай­ных, воз­му­щаю­щих сре­ду сил и/или при их при­сут­ствии.

Об­щий кри­те­рий воз­ник­но­ве­ния тур­бу­лент­но­сти уста­нов­лен Осбор­ном Рей­нольд­сом (1842–​1912) в 1883 го­ду. Он под­кра­ши­вал ла­ми­нар­ную струй­ку то­ка во вход­ной ча­сти стек­лян­ной труб­ки и сле­дил, ко­гда те­че­ние ста­нет тур­бу­лент­ным, фик­си­руя при этом кри­ти­че­ское зна­че­ние без­раз­мер­но­го опре­де­ляю­ще­го па­ра­мет­ра, на­зван­но­го впо­след­ствии в его честь чис­лом Рей­нольд­са. В Ман­че­с­тер­ском уни­вер­си­те­те, где Рей­нольдс про­во­дил свои опы­ты, со­хра­ни­лась его экс­пе­ри­мен­таль­ная уста­нов­ка.

По­рож­де­ние тур­бу­лент­но­сти ре­шет­кой

Ри­су­нок 1. По­рож­де­ние тур­бу­лент­но­сти ре­шет­кой. Ис­точ­ник

Пред­ста­вив ско­рость u_i в ви­де сум­мы сред­ней \bar u_i и пуль­са­ци­он­ной u_i^\prime со­став­ляю­щих, а дав­ле­ние в ви­де p=\bar p+p^\prime, Рей­нольдс по­лу­чил урав­не­ния для сред­них ве­ли­чин, но­ся­щие его имя:

\rho\frac{\partial\bar{u}_j\bar{u}_i}{\partial x_j} = \frac{\partial}{\partial x_j} 
\left[-\bar{p}\delta_{ij}+\mu\left(\frac{\partial\bar{u}_i}{\partial x_j}+\frac{\partial\bar{u}_j}{\partial x_i}\right)-
\rho\overline{u_i^\prime u_j^\prime}\right].
(7)

Здесь под­ра­зу­ме­ва­ет­ся сум­ми­ро­ва­ние по ин­дек­су j. По срав­не­нию с урав­не­ни­ем На­вье–​Сток­са (6) это урав­не­ние вклю­ча­ет до­пол­ни­тель­ные на­пря­же­ния — так на­зы­вае­мые на­пря­же­ния Рей­нольд­са \rho\overline{u_i^\prime u_j^\prime}. По­пыт­ки най­ти их вид из пер­вых прин­ци­пов фи­зи­ки ока­за­лись безуспеш­ны­ми, по­это­му урав­не­ние (7) ста­ло ба­зой для раз­ви­тия эм­пи­ри­че­ских тео­рий.

К не­до­опре­де­лен­ной си­сте­ме (7) сле­ду­ет до­ба­вить урав­не­ния для пуль­са­ций и пра­ви­ла осред­не­ния. По­лу­чен­ную та­ким об­ра­зом рас­ши­рен­ную си­сте­му урав­не­ний сле­ду­ет на­зы­вать си­сте­мой Рей­нольд­са. Ре­ше­ние этой си­сте­мы кро­ме сред­не­го и пуль­са­ци­он­но­го сла­га­е­мо­го со­дер­жит еще вол­но­вой член.

Л.В. Кел­лер и А.А. Фрид­ман да­ли ана­ли­ти­че­скую фор­му­ли­ров­ку про­бле­мы тур­бу­лент­но­сти, све­дя за­да­чу к бес­ко­неч­ной си­сте­ме урав­не­ний для ста­ти­сти­че­ских мо­мен­тов. Джеф­ф­ри Тей­лор ввёл в рас­смот­ре­ние кор­ре­ля­ци­он­ные функ­ции, а так­же впер­вые ввёл по­ня­тия об од­но­род­ной и изо­троп­ной тур­бу­лент­но­сти.

Лью­ис Фри Ричард­сон (1881–​1953) вы­ска­зал глу­бо­кие со­об­ра­же­ния о «кас­кад­ном про­цес­се» пе­ре­да­чи энер­гии по спек­тру от круп­но­мас­штаб­ных мод к мел­ко­мас­штаб­ным. Эта кар­ти­на раз­ви­той тур­бу­лент­но­сти изоб­ра­же­на Ричард­со­ном в сти­хо­тво­ре­нии, ко­то­рое вхо­дит во мно­гие учеб­ни­ки по тур­бу­лент­но­сти:

Big whorls have little whorls,
Which feed on their velocity;
Little whorls have smaller whorls,
And so on unto viscosity.

Ричард­сон лю­бил по­э­зию. Свою ин­ту­и­тив­ную тео­рию тур­бу­лент­но­сти он со­здал, вдох­нов­лён­ный наблю­де­ни­я­ми за эво­лю­ци­ей об­ла­ков.

Па­ра­диг­ма тур­бу­лент­но­сти озна­ме­но­ва­ла пре­вра­ще­ние гид­ро­ди­на­ми­ки из чи­сто тео­ре­ти­че­ской на­у­ки, из кол­лек­ции ори­ги­наль­ных, но ото­рван­ных от жиз­ни ре­ше­ний, в при­клад­ную на­у­ку. Раз­лив­шая­ся ре­ка стре­ми­лась вой­ти в бе­ре­га ак­ту­аль­ной по­лез­но­сти.

Осборн Рей­нольдс

Осборн Рей­нольдс (Osborne Reynolds; 1842–​1912), ан­глий­ский ин­же­нер и фи­зик. Ро­дил­ся в Бел­фа­сте в се­мье свя­щен­но­слу­жи­те­ля. С 18 лет ра­бо­тал в ме­ха­ни­че­ской ма­стер­ской, по­сту­пил в Кем­бридж­ский уни­вер­си­тет, где изу­чал ма­те­ма­ти­ку и ме­ха­ни­ку. Окон­чил уни­вер­си­тет в 1867 г. В 1868–​1905 гг. — про­фес­сор ка­фед­ры стро­и­тель­ной ме­ха­ни­ки Ман­че­с­тер­ско­го уни­вер­си­те­та. С 1888 г. воз­глав­лял Ви­твор­тов­скую ин­же­нер­ную ла­бо­ра­то­рию. Ра­бо­ты Рей­нольд­са по­свя­ще­ны ме­ха­ни­ке, гид­ро­ди­на­ми­ке, теп­ло­те, элек­три­че­ству, маг­не­тиз­му. В 1883 г. Рей­нольдс уста­но­вил, что ла­ми­нар­ное те­че­ние пе­ре­хо­дит в тур­бу­лент­ное, ко­гда вве­ден­ная им без­раз­мер­ная ве­ли­чи­на (чис­ло Рей­нольд­са) пре­вы­ша­ет кри­ти­че­ское зна­че­ние. Чис­ло Рей­нольд­са ши­ро­ко ис­поль­зу­ет­ся при ре­ше­нии за­дач гид­ро- и аэро­ди­на­ми­ки в слу­чае ма­лых и сред­них до­зву­ко­вых ско­ро­стей. По­дроб­нее

Наи­бо­лее важ­ным вкла­дом в тео­рию тур­бу­лент­но­сти счи­та­ют­ся 4 ра­бо­ты Ан­дрея Ни­ко­ла­е­ви­ча Кол­мо­го­ро­ва (1903–​1987), впер­вые опуб­ли­ко­ван­ные в 1941 го­ду. В со­во­куп­но­сти они так и на­зы­ва­ют­ся — «тео­рия К 41». Кол­мо­го­ров за­ни­мал­ся тур­бу­лент­но­стью при­мер­но пол­го­да, до на­ча­ла вой­ны, а за­тем, в свя­зи с тре­бо­ва­ни­я­ми воен­но­го вре­ме­ни, за­нял­ся бал­ли­сти­кой. Ещё 2 ра­бо­ты Кол­мо­го­ро­ва от­но­сят­ся к 1961–​1962 го­дам и на­зы­ва­ют­ся «тео­рия К 62».

Бы­ли ли пред­ше­ствен­ни­ки у Рей­нольд­са? Ра­зу­ме­ет­ся, каж­дый че­ло­век ви­дел тур­бу­лент­ность в ат­мо­сфе­ре, в ру­чье, в струе во­ды. Ха­ген ещё в 1839 го­ду уста­но­вил, что ха­рак­тер те­че­ния во­ды в тру­бе из­ме­ня­ет­ся с уве­ли­че­ни­ем на­по­ра. Он счи­тал, что при опре­де­лён­ных усло­ви­ях в по­то­ке воз­ни­ка­ют внут­рен­ние вих­ри, и это при­во­дит к по­вы­ше­нию со­про­тив­ле­ния, и, сле­до­ва­тель­но, к умень­ше­нию рас­хо­да во­ды.

Мас­сив­ным бло­ком в ос­но­ва­нии тео­рии тур­бу­лент­но­сти ле­жит тео­рия ве­ро­ят­но­стей, пре­крас­ные ис­то­ри­че­ские об­зо­ры ко­то­рой да­ны Ши­ряе­вым (1998) и Гне­ден­ко (2001). В тео­рии тур­бу­лент­но­сти есть тео­рия устой­чи­во­сти и пе­ре­хо­да ла­ми­нар­но­го те­че­ния в тур­бу­лент­ное. Рэ­лею в 1880 го­ду, В. Ор­ру и А. Зо­ммер­фель­ду в 1907–​1908 го­дах уда­лось ли­не­а­ри­зо­вать за­да­чу устой­чи­во­сти в част­ном слу­чае пло­ско­па­рал­лель­но­го дви­же­ния. Боль­шое зна­че­ние име­ет тео­рия устой­чи­во­сти по­гра­нич­но­го слоя и ла­ми­нар­но-​тур­бу­лент­но­го пе­ре­хо­да в по­гра­нич­ном слое (Бэт­че­лор, 2000). Сле­ду­ет упо­мя­нуть да­ле­ко про­дви­ну­тую тео­рию изо­троп­ной тур­бу­лент­но­сти (Бэт­че­лор, 1955).

В XIX ве­ке ос­нов­ной гид­ро­ди­на­ми­че­ской мо­де­лью яв­ля­лась мо­дель иде­аль­ной жид­ко­сти, в XIX–​XX ве­ках — мо­дель вяз­кой жид­ко­сти. В XXI ве­ке учё­ные ста­ли вплот­ную за­ни­мать­ся по­строе­ни­ем фи­зи­че­ских и ма­те­ма­ти­че­ских мо­де­лей тур­бу­лент­но­сти. Сле­ду­ет от­ме­тить, что па­ра­диг­ма тур­бу­лент­но­сти в гид­ро­ди­на­ми­ке еще не сфор­му­ли­ро­ва­на, по­ка вид­ны толь­ко её кон­ту­ры... По­строе­ны ма­те­ма­ти­че­ские мо­де­ли слу­чай­ных про­цес­сов в тео­рии бро­унов­ско­го дви­же­ния и в кван­то­вой ме­ха­ни­ке. Од­на­ко по­стро­ить адек­ват­ную мо­дель ис­то­рии че­ло­ве­че­ства, мак­ро­эко­но­ми­ки и тур­бу­лент­но­сти по­ка не уда­лось.

Не­воз­мож­но про­ве­сти чёт­кую гра­ни­цу меж­ду гид­ро­ди­на­ми­кой и не­ли­ней­ной ме­ха­ни­кой. И там, и там важ­на не­ли­ней­ность. Ак­ту­аль­но как изу­че­ние гид­ро­ди­на­ми­че­ских урав­не­ний (Бус­си­неска, Кор­те­ве­га–​де Ври­за и про­чих), так и ре­ше­ние урав­не­ний не­ли­ней­ной ме­ха­ни­ки (Шрё­дин­ге­ра, Гин­збур­га–​Лан­дау, Си­нус–​Гор­до­на и про­чих), ко­то­рые на­хо­дят важ­ные при­ме­не­ния в гид­ро­ди­на­ми­ке.

Вы­даю­щий­ся фи­зик Вер­нер Гей­зен­берг (1901–​1976), се­рьёз­но за­ни­мав­ший­ся про­бле­мой тур­бу­лент­но­сти, при­знал­ся на смерт­ном од­ре, что хо­тел бы за­дать Гос­по­ду два во­про­са — об ос­но­вах тео­рии от­но­си­тель­но­сти и о при­чи­не тур­бу­лент­но­сти. «Ду­маю, что Гос­подь от­ве­тит мне на пер­вый из них», — га­лант­но за­клю­чил он.

Чис­лен­ное мо­де­ли­ро­ва­ние гид­ро­ди­на­ми­че­ских про­цес­сов

Услож­не­ние ма­те­ма­ти­че­ских мо­де­лей при­ве­ло к то­му, что че­ло­век пе­ре­стал справ­лять­ся с рас­че­та­ми. От­ве­том на это яви­лось со­зда­ние ком­пью­те­ра. В то вре­мя ка­за­лось, что все за­да­чи бу­дут не­мед­лен­но ре­ше­ны, что вос­тор­же­ство­вал прин­цип пи­фа­го­рей­цев: «Всё есть чис­ло». Од­на­ко ко­нец эм­пи­ри­че­ской эпо­хи в на­у­ке и тех­ни­ке не на­сту­пил, ото­дви­нув­шись на сле­дую­щее сто­ле­тие. Со­вре­мен­ный ком­пью­тер поз­во­ля­ет ре­шать слож­ные за­да­чи гид­ро­ди­на­ми­ки, фи­зи­ки плаз­мы, ана­ли­за за­гряз­не­ния воз­ду­ха и грун­то­вых вод, со­зда­ния но­вых ле­карств, со­став­ле­ния карт озо­но­во­го слоя, сей­сми­че­ско­го ана­ли­за. Чис­лен­ные рас­чё­ты вос­тре­бо­ва­ны как в фун­да­мен­таль­ных на­у­ках, так и в при­клад­ных.

Для учё­ных и ин­же­не­ров чис­лен­ный рас­чёт стал обы­ден­ным и мощ­ным ору­жи­ем на­ступ­ле­ния на не­по­знан­ное. Чис­лен­ные рас­чё­ты прак­ти­че­ски вы­тес­ни­ли из упо­треб­ле­ния та­кие эм­пи­ри­че­ские ме­то­ды, как при­бли­же­ния Буб­но­ва–​Га­лёр­ки­на в тео­рии по­гра­нич­но­го слоя, ме­то­ды ти­па Бет­ца–​Мульт­хо­па в тео­рии кры­ла боль­шо­го удли­не­ния, по­лу­эм­пи­ри­че­ские ме­то­ды в тео­рии тур­бу­лент­но­сти и так да­лее. В на­сто­я­щее вре­мя раз­ра­бо­та­ны на­дёж­ные ме­то­ды рас­чё­та ла­ми­нар­ных те­че­ний, од­на­ко безэм­пи­ри­че­ский рас­чёт тур­бу­лент­ных те­че­ний по­ка ещё не­воз­мо­жен. Вы­чис­ли­тель­ная гид­ро­ди­на­ми­ка — ос­нов­ная часть вы­чис­ли­тель­ной ма­те­ма­ти­ки, ибо в ней, как ни в од­ной об­ла­сти фи­зи­ки, за­дей­ство­ва­на не­ли­ней­ность и, как след­ствие, тур­бу­лент­ность.

Ком­пью­тер­ная мо­дель жид­ко­сти пред­став­ля­ет со­бой при­бли­же­ние ма­те­ма­ти­че­ской мо­де­ли. Она пред­став­ля­ет со­бой в не­ко­то­рой сте­пе­ни «мо­дель мо­де­ли».

Рас­ска­жу вкрат­це об од­ном под­хо­де к чис­лен­но­му мо­де­ли­ро­ва­нию жид­ко­сти: ме­то­де ко­неч­ных эле­мен­тов. Для то­го что­бы хра­нить со­стоя­ние жид­ко­сти в па­мя­ти ком­пью­те­ра, имею­щей ко­неч­ный объ­ём, в жид­ко­сти вы­де­ля­ют ко­неч­ное чис­ло то­чек, па­ра­мет­ры в ко­то­рых вы­чис­ля­ют­ся. Счи­та­ет­ся, что каж­дая точ­ка не­по­сред­ствен­но вза­и­мо­дей­ству­ет лишь с не­сколь­ки­ми со­сед­ни­ми точ­ка­ми. Ес­ли со­еди­нить вза­и­мо­дей­ствую­щие точ­ки от­рез­ка­ми, то вы­чис­ля­е­мая об­ласть по­кро­ет­ся сет­кой (см. ри­су­нок 2). Та­кая сет­ка на­зы­ва­ет­ся рас­чёт­ной сет­кой, а вы­чис­ляе­мые точ­ки — уз­ла­ми этой сет­ки.

При­мер рас­чет­ной сет­ки для вы­чис­ле­ния те­че­ний в Азов­ском мо­ре

Ри­су­нок 2. При­мер рас­чет­ной сет­ки для вы­чис­ле­ния те­че­ний в Азов­ском мо­ре

Обыч­но за ос­но­ву бе­рут­ся урав­не­ния На­вье–​Сток­са. Они за­тем мо­дифи­ци­ру­ют­ся пу­тём вве­де­ния в рас­смот­ре­ние важ­ных фак­то­ров: глу­би­ны, гра­ви­та­ции, осад­ков, си­лы Ко­рио­лис­са, вет­ра, вяз­ко­сти, тур­бу­лент­ной диф­фу­зии, сто­ков рек,…

Па­ра­мет­ры жид­ко­сти (дав­ле­ние, ком­по­нен­ты ско­ро­сти, со­лё­ность, тем­пе­ра­ту­ра и дру­гие) в уз­лах сет­ки свя­зы­ва­ют­ся урав­не­ни­я­ми, ко­то­рые по­лу­ча­ют­ся ин­тег­ри­ро­ва­ни­ем ис­ход­ных диф­фе­рен­ци­аль­ных урав­не­ний по ячей­кам сет­ки. Об­ра­зу­ет­ся се­точ­ная си­сте­ма урав­не­ний.

Вре­мя то­же дро­бит­ся на от­рез­ки. Вы­во­дит­ся за­ви­си­мость меж­ду зна­че­ни­я­ми па­ра­мет­ров жид­ко­сти в сет­ке в те­ку­щий мо­мент вре­ме­ни и, на­при­мер, зна­че­ни­я­ми па­ра­мет­ров се­кун­ду на­зад (это на­зы­ва­ет­ся вре­мен­ным ша­гом). Та­ким об­ра­зом, се­кун­да за се­кун­дой, мы мо­жем рас­счи­тать год су­ще­ство­ва­ния во­до­ё­ма.

По­лу­ча­ет­ся, что па­ра­мет­ры каж­до­го уз­ла сет­ки на­хо­дят­ся в за­ви­си­мо­сти от окру­жаю­щих его уз­лов (с ко­то­ры­ми он свя­зан рёб­ра­ми). Уз­лы сет­ки как бы яв­ля­ют­ся боч­ка­ми с во­дой, а рёб­ра — тру­ба­ми, со­еди­няю­щи­ми эти боч­ки. Оста­ёт­ся толь­ко на­де­ять­ся, что по­доб­ная «сеть труб и бо­чек», бу­дучи до­ста­точ­но по­дроб­ной, даст при­бли­же­ние к те­че­нию жид­ко­сти в ре­аль­ном во­до­ё­ме.

По­сле на­пи­са­ния ком­пью­тер­ной про­грам­мы, она за­пус­ка­ет­ся на счёт для раз­лич­ных зна­че­ний па­ра­мет­ров. Каж­дое вы­чис­ле­ние кар­ты те­че­ний с ис­поль­зо­ва­ни­ем пер­со­наль­но­го ком­пью­те­ра мо­жет за­ни­мать не­сколь­ко ча­сов и да­же дней. Что­бы умень­шить вре­мя вы­чис­ле­ний со­зда­ют­ся па­ра­л­лель­ные про­грам­мы, спо­соб­ные ра­бо­тать од­но­вре­мен­но на де­сят­ках ком­пью­те­ров, объ­еди­няя их вы­чис­ли­тель­ные ре­сур­сы.

Ре­зуль­тат гид­ро­ди­на­ми­че­ско­го мо­де­ли­ро­ва­ния

Ри­су­нок 3. Ре­зуль­тат мо­де­ли­ро­ва­ния. Се­вер­ный ве­тер, 5 м/с.
Цве­том по­ка­за­но воз­вы­ше­ние уров­ня во­ды из-за вет­ра

И вот, на­ко­нец, по­сле мно­го­крат­но­го ис­прав­ле­ния про­грам­мы и до­ра­бот­ки мо­де­ли мы по­лу­ча­ем ре­зуль­та­ты, ко­то­рые счи­та­ем ито­го­вы­ми. Но как узнать, яв­ля­ют­ся ли эти ре­зуль­та­ты вер­ны­ми? Со­от­вет­ству­ют ли они дей­стви­тель­но­сти? Для это­го ис­поль­зу­ет­ся так на­зы­ва­е­мая ве­рифи­ка­ция мо­де­ли: мо­дель про­ве­ря­ет­ся на ис­ход­ных дан­ных, ре­зуль­тат для ко­то­рых из­ве­с­тен (из­ме­рен или точ­но вы­чис­лен). Это мо­жет быть, на­при­мер, дру­гая, про­стая фор­ма во­до­ё­ма с за­дан­ны­ми по­год­ны­ми усло­ви­я­ми, для ко­то­рой из­ве­ст­на кар­ти­на те­че­ний. И, ес­ли мо­дель (про­грам­ма) да­ла точ­ный ре­зуль­тат на из­ве­ст­ных дан­ных, счи­та­ет­ся, что она бу­дет да­вать при­ем­ле­мые ре­зуль­та­ты и на дру­гих, прак­ти­че­ски по­лез­ных дан­ных, ре­зуль­тат для ко­то­рых не­из­ве­с­тен. Это по­хо­же на сда­чу эк­за­ме­нов по во­жде­нию ав­то­мо­би­ля: ес­ли уче­ник зна­ет пра­ви­ла (про­грам­ма на­пи­са­на без оши­бок) и вы­пол­нил все упраж­не­ния (про­грам­ма да­ёт хо­ро­шие ре­зуль­та­ты на те­сто­вых за­да­чах), то счи­та­ет­ся, что в даль­ней­шем он бу­дет хо­ро­шо ез­дить на ав­то­мо­би­ле по го­ро­ду.

Од­на­ко чис­лен­ное мо­де­ли­ро­ва­ние те­че­ний жид­ко­сти не так про­сто и безоб­лач­но, как мо­жет по­ка­зать­ся. Вот, на мой взгляд, не­ко­то­рые про­бле­мы вы­чис­ли­тель­ной гид­ро­ди­на­ми­ки, над ре­ше­ни­ем ко­то­рых сей­час упор­но ра­бо­та­ют ис­сле­до­ва­те­ли:

  1. Пре­жде все­го, урав­не­ния На­вье–​Сток­са — не­ли­ней­ные. Воз­мож­но су­ще­ство­ва­ние не­сколь­ких ре­ше­ний, чис­лен­ные ме­то­ды до­ста­точ­но слож­ны, труд­но до­бить­ся га­ран­ти­ро­ван­ной схо­ди­мо­сти.
  2. Жид­кость — слож­ная суб­стан­ция, об­ла­даю­щая раз­лич­ны­ми ме­няю­щи­ми­ся во вре­ме­ни и про­стран­стве ха­рак­те­ри­сти­ка­ми. За­ча­стую по­лу­ча­ет­ся так, что при вы­чис­ле­нии од­ной ха­рак­те­ри­сти­ки мы на­ру­ша­ем дру­гую. Тре­бу­ют­ся вся­кие изощ­рён­ные ме­то­ды, что­бы это­го не слу­ча­лось (на­при­мер, по­прав­ка к дав­ле­нию).
  3. Нуж­но до­бить­ся вы­пол­не­ния за­ко­нов со­хра­не­ния энер­гии и им­пуль­са. Диф­фе­рен­ци­аль­ные урав­не­ния гид­ро­ди­на­ми­ки со­дер­жат эти за­ко­ны лишь кос­вен­но: ес­ли их про­ин­тег­ри­ро­вать по всей об­ла­сти, то по­лу­чит­ся, что сум­мар­ные энер­гия и им­пульс в ней со­хра­ня­ют­ся (не ме­ня­ют­ся со вре­ме­нем). Чис­лен­ные схе­мы, да­же ес­ли они не­пло­хо при­бли­жа­ют урав­не­ния (и те­че­ния) ло­каль­но, мо­гут да­вать су­ще­ствен­ные рас­хож­де­ния в та­ких гло­баль­ных па­ра­мет­рах, как энер­гия и им­пульс.
  4. Схем­ная вяз­кость: каж­дая ячей­ка/точ­ка/эле­мент сет­ки об­ла­да­ют не­ко­то­рым на­бо­ром па­ра­мет­ров. Но этот на­бор — ко­неч­ный. На­при­мер, мы мо­жем хра­нить сред­ние дав­ле­ние и ско­рость жид­ко­сти в ячей­ке. Это озна­ча­ет, что на каж­дом ша­ге по вре­ме­ни мы бу­дем упор­но те­рять все те осо­бен­но­сти те­че­ния, ко­то­рые не мо­гут быть опи­са­ны имею­щим­ся в ячей­ке (и в сет­ке в це­лом) на­бо­ром па­ра­мет­ров. В слу­чае хра­не­ния сред­них зна­че­ний, эти осо­бен­но­сти как бы мгно­вен­но «раз­ма­зы­ва­ют­ся» по всей ячей­ке. Это на­по­ми­на­ет до­пол­ни­тель­ную вяз­кость, ко­то­рая бу­дет за­ви­сеть от раз­ме­ров и фор­мы яче­ек сет­ки. Мо­де­ли­ро­ва­ние те­че­ния не­вяз­кой жид­ко­сти ста­но­вит­ся край­не слож­ным.
  5. Пло­хая устой­чи­вость: за­ча­стую при­хо­дит­ся ре­шать за­да­чу гид­ро­ди­на­ми­ки при очень ма­лых зна­че­ни­ях вре­мен­но­го ша­га, так как при боль­ших ша­гах схе­ма по­про­сту «раз­ва­ли­ва­ет­ся» (воз­ни­ка­ют на­рас­таю­щие ко­ле­ба­ния па­ра­мет­ров). Боль­шое чис­ло вре­мен­ных ша­гов при­во­дит к мед­лен­ным вы­чис­ле­ни­ям, на­коп­ле­нию по­греш­но­стей, и уси­ли­ва­ет схем­ную вяз­кость.
  6. Тур­бу­лент­ность: до сих пор не до кон­ца ис­сле­до­ван­ное яв­ле­ние. Ес­ли с рас­че­том ла­ми­нар­ных по­то­ков учё­ные на­учи­лись справ­лять­ся, то для ре­ше­ния за­дач с тур­бу­лент­но­стью каж­дый раз при­хо­дит­ся ду­мать. Тур­бу­лент­ность воз­ни­ка­ет рез­ко и на всех мас­шта­бах рас­стоя­ния. Воз­ни­ка­ют про­бле­мы учё­та мас­шта­бов тур­бу­лент­но­сти, срав­ни­мых с раз­ме­ром яче­ек и мень­ших их. Для это­го су­ще­ству­ют раз­лич­ные мо­де­ли, так или ина­че при­бли­жаю­щие тур­бу­лент­ность и её сред­ние ха­рак­те­ри­сти­ки.
  7. В при­клад­ных за­да­чах за­ча­стую тре­бу­ет­ся учесть до­пол­ни­тель­ные фак­то­ры, что зна­чи­тель­но услож­ня­ет мо­де­ли­ро­ва­ние: де­фор­ма­ция сте­нок тру­бо­про­во­да, тем­пе­ра­ту­ра, со­лё­ность, ре­льеф дна, вол­не­ние на по­верх­но­сти (по­движ­ная верх­няя гра­ни­ца), ве­тер, си­ла Ко­риол­ли­са, ис­па­ре­ние и осад­ки, при­ли­вы и от­ли­вы, под­зем­ные ис­точ­ни­ки и про­чее, и про­чее...
  8. А ещё бы­ва­ют сверх­зву­ко­вые те­че­ния, об­ра­зо­ва­ние пу­зырь­ков (раз­рыв по­то­ка), элек­тро­маг­нит­ные яв­ле­ния в жид­ко­сти при её те­че­нии в маг­нит­ном по­ле...

В за­клю­че­ние это­го пунк­та при­ве­ду ещё не­сколь­ко ком­пью­тер­ных ил­лю­стра­ций чис­лен­но­го мо­де­ли­ро­ва­ния те­че­ния жид­ко­сти:

Об­ра­зо­ва­ние вих­рей в вяз­кой жид­ко­сти

Ри­су­нок 4. Об­ра­зо­ва­ние вих­рей в вяз­кой жид­ко­сти воз­ле гра­ниц из од­но­го на­чаль­но­го вих­ря, сме­щён­но­го от­но­си­тель­но цен­тра об­ла­сти. По­ка­за­ны изо­ли­нии дав­ле­ния: дав­ле­ние внут­ри вих­ря все­гда ни­же, чем дав­ле­ние сна­ру­жи

Ис­те­че­ние жид­ко­сти из соп­ла

Ри­су­нок 5. Ис­те­че­ние жид­ко­сти из соп­ла

Ни­ти то­ка жид­ко­сти

Ри­су­нок 6. «Ни­ти то­ка» жид­ко­сти, про­те­каю­щей че­рез тру­бу слож­ной фор­мы

За­клю­че­ние

В дан­ной ра­бо­те не за­тро­ну­ты по­верх­ност­ное на­тя­же­ние жид­ко­сти, фа­зо­вые пе­ре­хо­ды и про­чие свой­ства жид­ко­сти, как хи­ми­че­ской и фи­зи­че­ской суб­стан­ции. Рас­смот­ре­но лишь те­че­ние (дви­же­ние) жид­ко­сти.

Кро­ме то­го, у са­мой зна­ко­мой нам жид­ко­сти — во­ды — есть са­мые уди­ви­тель­ные свой­ства из всех жид­ко­стей на Зем­ле. Очень ре­ко­мен­дую озна­ко­мить­ся с кни­гой [4].

На­де­юсь, что я смог за­ин­те­ре­со­вать чи­та­те­ля уди­ви­тель­ной на­у­кой гид­ро­ди­на­ми­кой. Ис­то­рия этой на­у­ки, оче­вид­но, не за­кон­че­на. Ещё мно­го сто­ле­тий она бу­дет за­ни­мать умы учё­ных. Кто зна­ет, воз­мож­но, в бу­ду­щем по­явят­ся про­стые урав­не­ния, опи­сы­ваю­щие всё мно­го­об­ра­зие фе­но­ме­нов, наблю­дае­мых при дви­же­нии во­ды. Но мо­жет слу­чить­ся и так, что урав­не­ния (и пред­став­ле­ния о жид­ко­сти) бу­дут услож­нять­ся, вклю­чая в се­бя всё но­вые и но­вые яв­ле­ния.

Спи­сок ис­поль­зо­ван­ных ма­те­ри­а­лов

По­сле но­ме­ра сто­ит про­цент, по­ка­зы­ваю­щий, ка­кая часть тек­ста взя­та из дан­но­го ис­точ­ни­ка.

  1. 30%. 11th Edition of the Enciclopaedia Britannica. 1911 г. На ан­глий­ском язы­ке. Осо­бен­но ин­те­рес­на ста­тья «Hydromechanics».
  2. 30%. Бе­тя­ев С. К. «Про­ле­го­ме­ны к ме­та­гид­ро­ди­на­ми­ке». РХД, 2006 г. 304 стр.
  3. 20%. Wikipedia, the Free Encyclopedia. На ан­глий­ском и рус­ском язы­ках.
  4. 0%. Лео­нид Адоль­фо­вич Куль­ский, Во­ля Ва­си­льев­на Даль, Люд­ми­ла Гри­го­рьев­на Лен­чи­на. «Во­да зна­ко­мая и за­га­доч­ная». Из­датель­ство «Ра­дянсь­ка шко­ла», 1982 г. 53 стр. Ссыл­ка

Осталь­ные 20% — соб­ствен­ный текст.

431 отзыв на «историю гидродинамики»

Огром­ное спа­си­бо за «Ис­то­рию гид­ро­ди­на­ми­ки». Моя дочь Ва­ше­го воз­рас­та в обо­зри­мом бу­ду­щем за­щи­ща­ет кан­ди­дат­скую по од­ной из тем тур­бу­лент­но­сти, а моя ин­же­нер­ная спе­ци­аль­ность от это­го да­ле­ка. Те­перь мне бу­дет лег­че най­ти об­щий язык с до­че­рью. Бле­стя­щая ста­тья, гид­ро­ди­на­ми­кой за­ин­те­ре­со­ва­ли!
Ан­тон, ве­ли­ко­леп­ный сайт. Не­дав­но на­чал тес­но за­ни­мать­ся с CFD-ком­плек­са­ми и ма­те­ма­ти­че­ским и ин­же­нер­ным ана­ли­зом слож­ных рас­че­тов — на ваш сайт на­ткнул­ся в по­ис­ке ин­фор­ма­ции для ре­фе­ра­та по ис­то­рии на­у­ки для ас­пи­ран­ту­ры — очень по­нра­ви­лась ста­тья ва­ша по гид­ро­ди­на­ми­ке, хо­чу ис­поль­зо­вать ее в сво­ей ра­бо­те — в спис­ке ис­точ­ни­ков пре­не­пре­мен­но ука­жу ваш сайт
Ста­тья по ис­то­рии гид­ро­ди­на­ми­ки — это и есть мой ре­фе­рат по ис­то­рии на­у­ки для ас­пи­ран­ту­ры. Вот та­кие сов­па­де­ния.
Ре­фе­рат по ис­то­рии на­у­ки для ас­пи­ран­ту­ры — ак­ту­аль­ная те­ма! Ан­тон, ес­ли Вы не воз­ра­жа­е­те, я то­же вос­поль­зу­юсь Ва­шим гра­мот­но обоб­щен­ным ма­те­ри­а­лом и то­же пре­не­пре­мен­но ука­жу сайт в спис­ке ис­точ­ни­ков. Рад ви­деть лю­дей с близ­ки­ми ин­те­ре­са­ми!
Мне то­же для ре­фе­ра­та для ас­пи­ран­ту­ры на кан­ди­дат­ский нуж­на ис­то­рия гид­ро­ди­на­ми­ки! Класс­но, спа­си­бо за ин­фор­ма­цию… Нас по­том всех не по­прут из ас­пи­ран­ту­ры за оди­на­ко­вые ре­фе­ра­ты?
Те­перь нас уже трое — то­же ре­фе­рат, то­же кан­ди­дат­ский!
Огра­ни­че­ние, что жид­кость не­сжи­ма­е­ма , от­бра­сы­ва­ет на­ли­чие в жид­ко­сти энер­гии сжа­тия (ко­эф­фи­ци­ент сжи­ма­е­мо­сти = 0) .
Но жид­кость про­дол­жа­ет иметь те­ку­честь (из­ме­ня­ет фор­му не из­ме­няя объ­ем) и про­дол­жа­ет течь по тру­бам .
При ис­те­че­нии жид­ко­сти че­рез от­вер­стие из со­су­да Тор­ри­чел­ли фи­зи­ки мо­гут наблю­дать не­пре­рыв­ное пре­об­ра­зо­ва­ние по­тен­ци­аль­ной энер­гии на­коп­лен­ной жид­ко­стью в со­су­де в по­ле си­лы тя­го­те­ния Зем­ли в ки­не­ти­че­скую ( то есть энер­гию дви­же­ния) .
По­сколь­ку жид­кость не­сжи­ма­е­ма , то при­об­ре­та­е­мая жид­ко­стью ки­не­ти­че­ская энер­гия мо­жет быть объ­яс­не­на толь­ко на ос­но­ве ди­на­ми­ки твер­до­го те­ла .
А имен­но всем из­вест­но­го вто­ро­го за­ко­на Нью­то­на.
Его фун­да­мент – из­ме­не­ние ско­ро­сти те­ла мас­сой М под дей­стви­ем си­лы
(то есть влия­ния на не­го дру­го­го те­ла).
У нас это влия­ние вы­ра­же­но в ви­де воз­дей­ствия од­ной ча­сти жид­ко­сти (од­но­го те­ла) на дру­гую ее часть (дру­гое те­ло) .
При вы­тал­ки­ва­нии ча­сти жид­ко­сти из со­су­да (пред­став­ля­ем се­бе твер­дое те­ло) , она за ко­неч­ное вре­мя при­об­ре­та­ет ко­неч­ную ско­рость дви­же­ния .
А зна­чит и ко­неч­ное воз­дей­ствие со сто­ро­ны жид­ко­сти на­хо­дя­щей­ся в со­су­де (ина­че бы ско­рость по­сто­ян­но уве­ли­чи­ва­лась, что про­ти­во­ре­чит тео­ре­ме Тор­ри­чел­ли ).
Та­ким об­ра­зом про­ис­хо­дит не толь­ко из­ме­не­ние ско­ро­сти под дей­стви­ем си­лы , но что не ме­нее важ­но — из­ме­не­ние си­лы воз­дей­ствия (от мак­си­маль­но­го зна­че­ния до ну­ле­во­го ) при из­ме­не­нии ско­ро­сти дви­же­ния вы­те­ка­е­мой из от­вер­стия жид­ко­сти.
А урав­не­ние Бер­нул­ли – след­ствие тео­ре­мы Тор­ри­чел­ли .
Нет со­су­да Тор­ри­чел­ли – не­при­ме­ни­мо урав­не­ние Бер­нул­ли .
Дви­же­ние сжа­той жид­ко­сти в за­мкну­той тру­бе раз­лич­ных се­че­ний не со­дер­жит со­су­да Тор­ри­чел­ли ( жид­кость име­ет ки­не­ти­че­скую энер­гию и энер­гию
сжа­тия ). Как при­ме­нить урав­не­ние Бер­нул­ли в этом слу­чае?
Ста­тья Ан­то­на по­зна­ва­тель­на. Спа­си­бо.
«Ис­то­рия га­зо­вой ме­ха­ни­ки из­ло­же­на Я.Б. Зель­до­ви­чем. Удар­ную вол­ну от­крыл «на кон­чи­ке пе­ра» Б. Ри­ман в 1876 го­ду. Од­на­ко ги­по­те­зу о су­ще­ство­ва­нии удар­ной вол­ны за­дол­го до не­го, в 1848 го­ду, вы­ска­зал Стокс, но от­ка­зал­ся от нее под влия­ни­ем кри­ти­ки В. Том­со­на и сво­е­го уче­ни­ка лор­да Рэ­лея.» Ри­ман ни­как не мог в 1876 го­ду от­крыть удар­ную вол­ну (он в 1866 го­ду умер). Это бы­ло 1860-м. Ис­правь­те, по­жа­луй­ста.
«Од­на­ко ги­по­те­зу о су­ще­ство­ва­нии удар­ной вол­ны за­дол­го до не­го, в 1848 го­ду, вы­ска­зал Стокс, но от­ка­зал­ся от нее под влия­ни­ем кри­ти­ки В. Том­со­на и сво­е­го уче­ни­ка лор­да Рэ­лея.» Здесь ошиб­ка пе­ре­во­да. В 1848 го­ду Стокс вы­ска­зал ги­по­те­зу о су­ще­ство­ва­нии удар­ной вол­ны, но от­ка­зал­ся от сво­ей ра­бо­ты. Од­на­ко Лорд Кель­вин и Лорд Рэ­лей поз­же убе­ди­ли его что он был всё же прав. Как то так. Ис­точ­ник http://www.potto.org/gasDynamics/node44.html
Ну ни­как в 1848 го­ду Рэ­лей ни­как не мог убе­дить Сток­са (Рэ­лею то­гда бы­ло 8 лет)…
Во мно­гих ме­стах ста­тья — пе­ре­сказ ан­гло­языч­ной «Ви­ки­пе­дии»: http://en.wikipedia.org/wiki/History_of_fluid_mechanics,
что вид­но уже из то­го, что Иоганн Бер­нул­ли (имен­но так при­ня­то в боль­шей ча­сти рус­ско­языч­ных ис­точ­ни­ков) на­зы­ва­ет­ся «Джо­ном Бер­нул­ли».
P.S. Не посмот­рел на вто­рую стра­ни­цу, где ав­тор чест­но рас­ска­зы­ва­ет об ис­точ­ни­ках. Про­шу про­ще­ния. Ссыл­ка на «Бри­тан­ни­ку» не ра­бо­та­ет, сайт за­крыл­ся. Спа­си­бо за ин­те­рес­ный сайт, с ин­те­ре­сом про­чи­тал дис­сер­та­цию ав­то­ра. Очень по­ра­до­ва­ли ав­то­ма­ти­че­ские пе­ре­но­сы и за­ме­ны ка­вы­чек на сай­те. Уда­чи.
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