Utafiti huu unachunguza mwitikio wa awamu moja wa maji ndani ya mfereji wenye mtiririko wa vurugu dhidi ya flux ya joto ya ukutani inayoongezeka kwa namna ya kielelezo, hadi mwanzo wa uchemkaji wa kinyuklia, kwa kutumia uigaji wa large eddy wenye azimio la juu. Lengo kuu la utafiti ni kuonyesha jinsi safu ya mpaka wa joto inavyojipanga upya wakati wa ongezeko la haraka la nguvu, si kwa upande wa joto pekee bali pia kwa upande wa uwanja wa kasi, miundo ya vortex, mikazo ya mnato na usafirishaji wa joto wa kimwelekeo.

Katika uigaji, joto halijachukuliwa kama scalar isiyotenda bali kama scalar hai inayobadilisha dynamic viscosity na thermal conductivity ya maji. Kwa njia hii, uwanja wa joto huathiri sifa za mtiririko, na uwanja wa mtiririko pia huathiri usambazaji wa joto kwa wakati mmoja. Joto la ukuta lililohesabiwa lililingana kwa ujumla na vipimo vya majaribio kabla ya kuanza kwa uchemkaji, kwa tofauti ya chini ya asilimia 5.

Kulingana na matokeo ya utafiti, upashaji wa haraka hupunguza mnato karibu na ukuta wa moto, hivyo kuongeza kasi katika mwelekeo wa mtiririko, kuimarisha vorticity katika mwelekeo wa upana, na kuunda katika safu ya buffer, takribani kati ya [ y^+=60\text{–}140 ], wasifu wa kasi karibu wa mstari unaofanana na mtiririko wa Couette. Miundo ya mtiririko iliyopashwa joto husongwa katika mwelekeo wa kawaida kwa ukuta huku ikipanuka katika mwelekeo wa upana wa mfereji. Mabadiliko haya huchangia kuhifadhi joto karibu na ukuta.

Katika mpito wa haraka, namba ya Stanton huongezeka kwa takribani asilimia 30, na eneo lililokua kikamilifu kwa upande wa joto hujitokeza mapema zaidi ndani ya mfereji. Kinyume chake, katika upashaji wa polepole, wa karibu na hali thabiti, namba ya Stanton hubaki ndani ya takribani asilimia 8 katika mpito wote, na eneo la kuingia kwa joto hudumu kwa muda mrefu zaidi.

Mara moja kabla ya mwanzo wa uchemkaji, upashaji wa haraka huunda safu pana na takribani sawia ya kioevu kilichopashwa kupita kiwango cha kuchemka kando ya ukuta. Katika upashaji wa karibu na hali thabiti, kioevu hicho kilichopashwa kupita kiasi hubaki ndani ya mikanda ya mwelekeo wa mtiririko inayohusiana na mistari ya vurugu yenye kasi ndogo. Hivyo muda wa upashaji huamua si tu kiwango cha jumla cha uhamishaji joto, bali pia mahali na mpangilio wa anga ambapo viputo vya kwanza vinaweza kujitokeza.

Utafiti haumodeli moja kwa moja kutokea kwa viputo wala uchemkaji wa awamu mbili. Hesabu zote zimewekewa kikomo katika hali ya kioevu ya awamu moja hadi wakati wa [ t^*=0 ], ambao ni wakati ishara ya kwanza ya uchemkaji wa kinyuklia ilipoonekana katika majaribio.

Tatizo linaloshughulikiwa na utafiti

Kuongezeka kwa haraka kwa joto linalopelekwa kwenye ukuta katika kipimo cha millisecond ni tatizo muhimu la usalama wa joto kwa reactors za nyuklia, mifumo ya elektroniki ya nguvu kubwa, injini za roketi na vipengele vinavyokabiliana na plasma katika reactors za fusion za baadaye. Mzigo wa joto unapoongezeka kwa kasi ya kutosha, kioevu kilicho karibu na ukuta kinaweza kupita joto la saturation hata wakati sehemu nyingine ya mfereji bado imepozwa sana.

Katika hali hizi mfumo unaweza kuelekea katika njia tatu tofauti:

• Upozaji unaweza kubaki katika hali ya forced convection ya awamu moja.
• Uchemkaji wa kinyuklia unaweza kuanza na kuongeza uhamishaji joto kwa muda.
• Katika hali kali zaidi, filamu ya mvuke inaweza kuundwa na kupunguza mgusano kati ya ukuta na kioevu, hivyo kupunguza kwa hatari uwezo wa kuondoa joto.

Hasa katika reactors za utafiti, ajali ya kuongezeka kwa reactivity inaweza kusababisha ongezeko takribani la kielelezo la nguvu ya fuel. Jinsi safu ya mpaka wa awamu moja ya coolant inavyoitikia ongezeko hili la haraka huathiri joto linalofikiwa na fuel cladding na wakati ambao uchemkaji huanza.

Masomo ya awali yalitegemea zaidi vipimo vilivyojumlishwa kama joto la wastani la ukuta, wastani wa coefficient ya uhamishaji joto au flux ya jumla ya joto. Vipimo hivyo havionyeshi moja kwa moja nyanja za kasi, vorticity, joto na stress za pande tatu zinazojitokeza ndani ya safu karibu na ukuta. Utafiti huu unalenga kuziba pengo hilo kwa kutumia uigaji wa large eddy wenye azimio la juu.

Maswali ya utafiti

Utafiti unashughulikia maswali makuu yafuatayo:

• Flux ya joto ya ukutani inayoongezeka kwa kielelezo inabadilishaje muundo wa karibu na ukuta wa mtiririko wa vurugu?
• Viscosity na thermal conductivity zinazotegemea joto zinaundaje muunganiko wa pande mbili kati ya momentum na uhamishaji joto?
• Kuna tofauti gani kati ya mpito wa haraka na mpito wa karibu na hali thabiti katika thermal entrance length, namba ya Stanton na usambazaji wa joto?
• Mara moja kabla ya kuanza kwa uchemkaji, kioevu kilichopashwa kupita kiasi huunda filamu sawia au mikanda ya vurugu?
• Flux za joto katika mwelekeo wa kawaida kwa ukuta, mwelekeo wa mtiririko na mwelekeo wa upana wa mfereji zinafuata mpangilio gani wa ukubwa?
• Je, wasifu wa joto katika nyakati na geometri tofauti za mfereji unaweza kuwakilishwa kwa uhusiano mmoja wa hisabati?

Mfumo wa majaribio na hali za uendeshaji

Modeli ya namba inategemea hali za majaribio za mzunguko wa mtiririko uliopunguzwa kwa kutumia maji yaliyodeionishwa. Geometri mbili nyembamba za mfereji wa mstatili zilichunguzwa:

Shinikizo la uendeshaji lilihifadhiwa 10 bar na joto la kuingia 353 K, yaani 80 °C. Kwa hali hizi, joto la saturation ni takribani 453 K, kwa hiyo kiwango cha subcooling cha kuingia kinafafanuliwa kama:

\[ \Delta T_{\mathrm{sub}}=T_{\mathrm{sat}}-T_{\mathrm{in}}=100\ \mathrm{K} \]

.

Kipengele cha kupasha joto kina urefu wa 20 mm na upana wa 4 mm. Filamu ya titanium yenye ukwaruzo wa kiwango cha nanometer iliwekwa juu ya substrate ya sapphire kwa mbinu ya vapor deposition. Uwezo wa sapphire kupitisha kwa sehemu mionzi ya infrared uliwezesha joto la uso wa heater kupimwa kutoka upande wa chini kwa infrared thermography.

Experimental Joule power na net wall heat flux inayopita kwenye kioevu si kitu kimoja. Kwa sababu substrate ya sapphire huhifadhi kiasi fulani cha nishati, hasa katika mpito wa [ \tau=5\ \mathrm{ms} ], net heat flux inayopitishwa kwenye kioevu haifuati kikamilifu curve ya kielelezo. Kama boundary condition ya namba, si electric power iliyorahisishwa bali net heat flux inayotegemea muda iliyopatikana kutokana na majaribio ndiyo iliyotumika.

Flux ya joto ya kielelezo na vigezo vya muda

Tabia ya jumla ya wall heat flux iliyowekwa ni:

\[ q''_w(t)\propto \exp\left(\frac{t}{\tau}\right) \]

. Hapa [ \tau ] ni characteristic time ya ongezeko la nguvu. Thamani ndogo ya [ \tau ] inamaanisha ongezeko la haraka zaidi la nguvu, na thamani kubwa inamaanisha ongezeko la polepole zaidi.

Kwa kulinganisha, joto la ukuta na muda vimefanywa dimensionless:

\[ T_w^*=\frac{T_w-T_{\mathrm{sat}}}{\Delta T_{\mathrm{sub}}} \]

\[ t^*=\frac{t-t_{\mathrm{ONB}}}{\tau} \]

\[ t_{\mathrm{ONB}} ] ni wakati ambapo uchemkaji wa kinyuklia huanza katika majaribio. Kwa hiyo:

• [ t^*<0 ] inawakilisha kipindi cha awamu moja kabla ya uchemkaji,
• [ t^*=0 ] inawakilisha mwanzo wa uchemkaji katika majaribio,
• [ t^*>0 ] inawakilisha kipindi cha awamu mbili kilicho nje ya upeo wa kimwili wa uigaji.

Mpangilio wa namba

Kielelezo 2.1 kinaonyesha maeneo mawili ya hesabu yanayofuatana katika mwelekeo wa mtiririko. Sehemu ya kwanza ni hydraulic recirculation box inayotumika kuzalisha turbulence iliyokua kikamilifu. Utoaji wa box hii hupelekwa kwenye inlet ya heated test section katika kila time step.

Katika test section:

• Wall heat flux inayotegemea muda na iliyo sawia kwa nafasi inawekwa kwenye ukuta wa chini.
• Ukuta wa juu ni adiabatic.
• No-slip condition inatumika kwenye kuta zote mbili.
• Periodic boundary condition inatumika katika mwelekeo wa upana wa mfereji.
• Outlet inahifadhiwa kwenye operating pressure ya 10 bar.
• Inlet fluid ina joto la 353 K.

Precursor simulations ziliendeshwa kwa 40–100 washout times hadi first-order turbulence statistics zikawa steady:

[ \tau_{\mathrm{wash}}=\frac{L_x}{U} \]

. Hapa [ L_x ] ni urefu wa computational domain katika mwelekeo wa mtiririko na [ U ] ni mean flow velocity.

Equations za conservation zilizotatuliwa

Mtiririko ulidhaniwa incompressible. Continuity equation:

\[ \nabla\cdot\mathbf{u}=0 \]

Momentum equation:

\[ \frac{\partial\mathbf{u}}{\partial t} +\nabla\cdot(\mathbf{u}\mathbf{u}) = -\frac{1}{\rho}\nabla P +\nabla\cdot\left[\nu(T)\nabla\mathbf{u}\right] -\nabla\cdot\boldsymbol{\tau}^{\mathrm{SGS}} \]

na temperature equation:

\[ \frac{\partial T}{\partial t} +\mathbf{u}\cdot\nabla T = \frac{1}{\rho c_p} \nabla\cdot\left[k(T)\nabla T\right] + \nabla\cdot\left(\alpha_t\nabla T\right) \]

zilitumika.

\[ \nu(T) ] inaonyesha kinematic viscosity inayotegemea joto, [ k(T) ] thermal conductivity inayotegemea joto, [ \boldsymbol{\tau}^{\mathrm{SGS}} ] stress ya small scales ambazo hazijatatuliwa, na [ \alpha_t ] subgrid-scale thermal diffusivity.

Density [ \rho ] na specific heat [ c_p ] zilikubaliwa kuwa constant. Density ilitathminiwa kwenye inlet temperature, na specific heat kwenye saturation temperature. Approach hii si fully compressible model yenye all thermophysical properties zinazotegemea joto.

Athari ya gravity ilipuuzwa. Kwa bulk Richardson number iliyohesabiwa:

[ Ri=\frac{(\Delta\rho/\rho)gL}{U^2} \]

kwa kuwa [ Ri<0{,}01 ], forced convection ilichukuliwa kutawala buoyancy.

Sifa za maji zinazotegemea joto

Sifa za maji zilichukuliwa kutoka CATHARE thermal-hydraulic property library. Kielelezo 2.2 kinaonyesha jinsi viscosity, thermal conductivity na Prandtl number zinavyobadilika na joto kwenye 10 bar.

Dynamic viscosity

Dynamic viscosity hupungua takribani kwa kielelezo joto linapoongezeka. Katika temperature range iliyofunikwa na uigaji, viscosity hupungua takribani mara tatu. Kupungua huku hubadilisha kwa kiasi kikubwa momentum diffusion na viscous stresses karibu na ukuta.

Thermal conductivity

Thermal conductivity si monotonic. Hubadilika ndani ya takribani asilimia 10 na kufikia maximum karibu 416 K. Katika superheated film iliyo karibu zaidi na ukuta, conductivity hupungua takribani asilimia 5. Kwa sababu joto kutoka ukutani huingia kwanza kwenye kioevu kwa conduction, hata kupungua huku kwa kiasi kidogo kuna umuhimu katika hatua ya mwisho ya mpito.

Prandtl number

Molecular Prandtl number imefafanuliwa kama:

\[ Pr=\frac{\mu c_p}{k} \]

. Wakati [ T>443\ \mathrm{K} ], [ Pr<1 ]. Hivyo katika cold core ya mfereji smallest thermal scale inahusishwa na Batchelor scale, na karibu na hot wall inahusishwa na Kolmogorov scale.

Kielelezo 3.4 kinaonyesha wall-normal–spanwise sections za thermal conductivity, viscosity na Prandtl number. Mabadiliko ya viscosity yanaenea hadi eneo la [ y^+>200 ], ilhali mabadiliko makubwa zaidi ya conductivity yanabaki katika eneo nyembamba zaidi karibu na ukuta.

Large eddy simulation na computational resolution

Hesabu zilifanywa kwa software ya TRUST/TrioCFD na resources za supercomputer ya TGCC ya Ufaransa. Cases ndogo zaidi zilitumia 10–40 million cells, na case kubwa zaidi takribani 650 million cells katika kila moja ya computational sections mbili.

Main numerical methods ni:

Near-wall grid spacing imepewa takribani [ \Delta y_w^+=0{,}5 ], streamwise spacing [ \Delta x^+=8 ] na spanwise spacing [ \Delta z^+=4 ]. Physical wall-normal cell height ni takribani 0,9–2,13 µm kulingana na condition.

WALE model na [ C_w=0{,}5 ] zilitumika kwa small-scale momentum transfer. Kwa thermal subgrid model turbulent Prandtl number:

\[ Pr_t=\frac{\nu_t}{\alpha_t}=0{,}9 \]

ilichaguliwa.

Katika quality assessment ya Appendix B, sehemu ya subgrid viscosity katika total effect ilibaki chini ya asilimia 10, na sehemu ya modeled thermal flux katika total thermal transfer ilibaki chini ya asilimia 1,2. Matokeo haya yanaonyesha kuwa sehemu kubwa ya near-wall mechanisms ilitatuliwa moja kwa moja. Hata hivyo, calculations si direct numerical simulation; ni LES.

Validation ya wall temperature

Kielelezo 3.1 kinalinganisha dimensionless wall temperatures kwa [ \tau=5 ], 20 na 100 ms na experimental results. Kabla ya kuanza kwa uchemkaji, simulation na experiment curves kwa ujumla ziko karibu.

Katika kipindi cha [ t^*<-1 ], tofauti kubwa zaidi inaonekana katika fast transitions. Researchers wanahusisha hii na assumption katika standard eddy-viscosity LES models ya local balance kati ya turbulence production na dissipation. Katika very rapid nonequilibrium flows, assumption hii inaweza kudhoofika.

Katika range ya [ -1<t^*<0 ], tofauti inapungua. Katika baadhi ya time steps mara moja kabla ya kuanza kwa uchemkaji, experimental na numerical temperatures karibu zinaingiliana.

Katika single-phase period, calculated wall temperatures kwa ujumla ziko slightly above experimental values. Hii inaonyesha simulation inaunderpredict heat-transfer coefficient kidogo. Tofauti kwa kawaida hubaki chini ya asilimia 5.

Wakati [ t^*>0 ], bubbles hujitokeza katika experiment wakati model inabaki single-phase, hivyo error inaongezeka haraka. Kwa hiyo study haitoi physical results kwa baada ya [ t^*=0 ].

Temperature overshoot at boiling onset

Rapid heat-flux increases huruhusu wall temperature kupita saturation temperature kwa kiwango kikubwa zaidi kabla ya nucleate boiling kuanza. Validation curves zinaripoti takribani 20 K wall superheat kwa quasi-steady transition na takribani 65 K kwa fast transition.

Katika mean wall-normal profiles kwenye thermally developed downstream region, wakati wa [ t^*=0 ] takribani:

\[ T^*\approx0{,}75\quad(\tau=5\ \mathrm{ms}) \]

\[ T^*\approx0{,}30\quad(\tau=100\ \mathrm{ms}) \]

zinaonekana. Kwa kuwa [ \Delta T_{\mathrm{sub}}=100\ \mathrm{K} ], hizi zinawakilisha takribani 75 K na 30 K local/subregion-averaged superheat. Values hizi hazina same spatial average kama validation averages za entire surface; kwa hiyo hazipaswi kutumika interchangeably.

Instantaneous temperature fields na turbulent streaks

Kielelezo 3.2 kinaonyesha temperature na local temperature-rate-of-change fields kwenye plane ya [ y^+=13 ]. Hot regions ni thin streaks zilizorefushwa katika streamwise direction. Spanwise spacing yao ni takribani 60–120 viscous units. Spacing hii inaendana na characteristic spacing inayojulikana ya near-wall low-speed turbulent streaks.

Katika takribani first [ 1000\delta_\nu ] distance kutoka inlet, joto hubaki mostly within viscous sublayer. Downstream zaidi, joto hufikia buffer layer, hot fluid ejects away from wall na kuchanganyika na large eddies kutoka cold core flow.

Katika temperature-rate-of-change field, neighboring heating na cooling regions zinaonekana. Folding na stretching ya interface kati ya hot na cold fluid na turbulent eddies huongeza thermal gradients na local energy transfer.

Mabadiliko ya vortex structures

Kielelezo 3.3 kinabainisha vortex cores kwa [ \lambda_2 ] criterion. Katika early transition, dense vortex structures za different sizes zinazoenea kutoka wall hadi channel core zinaonekana.

Wakati wa [ t^*=0 ], katika both fast na quasi-steady heating:

• Height ya wall-attached vortices inapungua,
• Density na length ya vortices inapungua,
• Structures zinacompressiwa kwenye thinner near-wall region.

Study inahusisha hii na viscosity decrease wakati temperature inaongezeka na reorganization ya near-wall turbulent kinetic energy.

Katika fast transition, wakati wa [ t^*=0 ], karibu all fluid immediately above heated surface inazidi saturation temperature. Katika quasi-steady transition, superheated regions zinabaki kama streamwise bands along low-speed streaks, zikiwa zimetenganishwa na cooler fluid.

Images hizi zinapendekeza two different boiling-onset pathways:

• Fast transition: Boiling onset iliyoenea across surface na spatially more simultaneous.
• Quasi-steady transition: Boiling inayohusiana na turbulent streaks, katika bands na kuanza downstream kwanza.

Kuongezeka kwa near-wall velocity

Dimensionless velocity:

\[ U^+=\frac{U_x}{u_\tau} \]

na wall coordinate:

\[ y^+=\frac{y}{\delta_\nu} \]

zilitumika. Katika classical isothermal viscous sublayer, approximately linear relation [ U^+=y^+ ] inatarajiwa.

Kulingana na Kielelezo 3.5, wakati wa [ t^*=0 ], slope ya near-wall velocity profile inaongezeka takribani asilimia 30 relative to canonical value. Katika same region magnitude ya dimensionless spanwise vorticity inaongezeka takribani mara mbili. Effect ni more pronounced katika [ \tau=5\ \mathrm{ms} ] transition.

Kwa sababu ya relation kati ya vorticity na velocity gradient, finding hii inaonyesha formation ya stronger shear layer karibu na hot wall.

Couette-like intermediate layer

Katika range ya [ y^+=60\text{–}140 ], velocity profile inakuwa almost perfectly linear. Reported coefficient of determination kwa linear fit ni:

\[ R^2>0{,}99 \]

. Katika same range, spanwise vorticity inaunda approximately constant plateau.

Constant velocity gradient na approximately constant vorticity zinaonyesha intermediate layer inayofanana na constant-shear Couette flow. Layer hii si independent Couette flow kati ya classical two plates; ni local similarity inayojitokeza katika buffer region ya turbulent channel kutokana na heating na viscosity variation.

Viscous shear stress

Total viscous stress katika LES framework imefafanuliwa kama:

\[ \tau_{ij} = \left[\mu(T)+\mu_t\right] \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \]

.

Main near-wall shear stress [ \tau_{xy} ] inapungua kadri transition inavyoendelea kutokana na viscosity decrease. Kwenye channel center inaendelea kubadilisha sign na zero point inabaki kwenye centerline.

Katika fast transition, [ \tau_{xy} ] profile inaunda clear plateau katika [ y^+\approx60\text{–}140 ]. Plateau hii inatokea katika same region na Couette-like layer iliyobainishwa kutoka velocity na vorticity profiles. Similar plateau haionekani kwenye unheated opposite wall, jambo linalo-support kwamba effect inahusiana na thermal-property variation badala ya general instantaneous turbulence fluctuation.

Normal stresses: wall-normal compression na lateral spreading

Kielelezo 3.7 kinachunguza normal stress components [ \tau_{yy} ] na [ \tau_{zz} ]. Magnitudes za components hizi ni takribani orders mbili chini ya main [ \tau_{xy} ] shear stress.

Kadri transition inavyoendelea:

• [ \tau_{yy} ] inakuwa more negative, ikionyesha wall-normal compression,
• [ \tau_{zz} ] inakuwa more positive, ikionyesha spanwise spreading.

Initially, [ \tau_{yy} ] extremum iko approximately [ y^+=20 ], na [ \tau_{zz} ] extremum approximately [ y^+=5 ]. Wakati wa [ t^*=0 ], two characteristic locations hizi zinakaribia approximately [ y^+=10 ] na [ y^+=9 ].

Stress pattern hii ina-support kwamba hot spots zinazotokea katika low-speed streaks zinakuwa thinner katika wall-normal direction huku zikirefuka spanwise. Researchers wanasema hii inasaidia kuhifadhi joto karibu na ukuta na kupunguza effective removal kuelekea core.

Wall-normal temperature profiles

Joto la fluid limefanywa dimensionless kama:

\[ T^*=\frac{T-T_{\mathrm{sat}}}{\Delta T_{\mathrm{sub}}} \]

. Subregions tatu— inlet, middle na outlet—zimelinganishwa kando ya mfereji.

Katika all times, temperature increase inabaki mostly within [ y^+\lesssim100 ], huku channel core ikibaki karibu na inlet temperature. Kwa hiyo ndani ya short transition period, si total fluid mass bali mainly thin near-wall layer ndiyo inapashwa joto.

Katika fast transition, inlet, middle na outlet profiles ziko closer to one another. Katika quasi-steady transition, downstream temperature development ni more pronounced na thermal entrance region ni longer.

Wakati wa [ t^*=0 ], fast transition inaunda near-wall layer above saturation temperature along entire heated surface, wakati katika quasi-steady transition sehemu ya inlet region bado iko below saturation temperature.

Stanton number na heat-transfer intensity

Local Stanton number imefafanuliwa kama:

\[ St(x^*,t^*) = \frac{q''_w(t^*)} {\rho c_pU\left[T_w(x^*,t^*)-T_{\mathrm{in}}\right]} \]

.

Kielelezo 3.9 kinaonyesha Stanton number ikiwa highest at inlet, kisha ikipungua downstream na kukaribia plateau. Plateau inaonyesha region ambapo thermal boundary layer imekuwa fully developed katika streamwise direction.

Quasi-steady transition

Katika [ \tau=100\ \mathrm{ms} ] case, Stanton number inabadilika only within approximately asilimia 8 throughout transition. Hii inaonyesha local heat-transfer coefficient kwa kiasi kikubwa inajirekebisha na increase ya wall loading na flow inabaki karibu na quasi-steady convection regime.

Fast transition

Katika [ \tau=5\ \mathrm{ms} ] case, Stanton number inaongezeka approximately asilimia 30 between [ t^*=-2 ] na [ t^*=-1 ] along entire heated length, kisha inabaki close to this high level until boiling onset.

Increase hii haimaanishi heat transfer ni weak katika fast transition. Kinyume chake, total transfer inaongezeka; lakini heat-rise rate ni kubwa kuliko rate ambayo entire turbulent velocity field inaweza kujirekebisha, kwa hiyo transient conduction na local energy storage zinakuwa more important.

Thermal entrance length

Streamwise direction imefanywa dimensionless kwa channel half-height:

\[ x^*=\frac{x}{e} \]

.

Katika fast transition, Stanton plateau inafikiwa approximately at:

\[ x^*\approx2\pi \]

. Katika quasi-steady transition, thermally fully developed state inapatikana only around:

\[ x^*\approx5\pi \]

near test-section outlet.

Shorter thermal entrance length katika rapid heating inahusishwa na surface response to sudden loading kupitia local energy storage na transient conduction. Katika slow heating, turbulent convection ina more time ku-develop katika streamwise direction na thermal structure inaenea farther.

Two-parameter sigmoid temperature law

Katika thermally fully developed region, temperature profiles zime-rescale kwa variable:

\[ \Theta=\frac{T-T_w}{T_{\mathrm{in}}-T_w} \]

. [ \Theta=0 ] corresponds to wall temperature na [ \Theta=1 ] corresponds to inlet temperature.

Profiles zinaonekana kufit two-parameter sigmoid relation katika logarithmic wall coordinate:

\[ \Theta(y^+) = \frac{(y^+)^k} {(y_0^+)^k+(y^+)^k} \]

Equivalent logistic representation:

\[ \Theta(y^+) = \frac{1} {1+\exp\left[-k\left(\ln y^+-\ln y_0^+\right)\right]} \]

.

\[ y_0^+ ] ni transition midpoint ambapo [ \Theta=1/2 ], na [ k ] inaonyesha transition sharpness katika logarithmic coordinate. Larger [ k ] ina maana sharper na more wall-confined layer, smaller [ k ] ina maana broader thermal layer.

TS1 fit parameters

TS2 fit parameters

All fits zilitoa [ R^2>0{,}99 ]. Fast transitions kwa ujumla zinazalisha smaller [ y_0^+ ] na larger [ k ]. Hii inathibitisha quantitatively kwamba thermal transition katika fast heating iko closer to wall na ni sharper.

Movement ya [ y_0^+ ] kwanza outward na kisha back toward wall near boiling onset inaendana na wall-normal compression iliyoonekana katika normal-stress analysis.

Local Péclet indicator

Local convection na conduction magnitudes zimelinganishwa kwa:

[ Pe= \frac{\left|\rho c_p\mathbf{u}\cdot\nabla T\right|} {\left|\nabla\cdot(k\nabla T)\right|} \]

. Ili kubadilisha very large [ Pe ] values kwenda limited range:

\[ \xi=\frac{Pe}{1+Pe} \]

imefafanuliwa.

• [ \xi=0 ] means conduction only,
• [ \xi=1/2 ] means equal conduction and convection,
• [ \xi=1 ] means convection dominance.

Kielelezo 3.11 kinaonyesha superheated regions wakati wa [ t^*=-1 ] mostly kama streamwise, conduction-dominated streaks. Wakati wa [ t^*=0 ], downstream ya thermal entrance region, most fluid exceeds saturation temperature; small conduction-dominated, mixed na convection-dominated regions coexist ndani ya same film.

Superheated film hii ni metastable. Study inapendekeza kwamba disturbance au nucleation site inayofaa ikitokea, inaweza kuwa prone to bubble formation over a broad surface area. Bubble nucleation haipo katika computational model.

Wall-normal regions za heat equation

Storage, conduction na convection terms zime-scale kwa ratio ya instantaneous wall heat flux to channel half-height:

\[ \frac{q''_w}{e} \]

. Kulingana na Kielelezo 3.12, thermal boundary layer inaweza kugawanywa approximately into three regions:

Katika fast transition, extrema za operators zinatokea at smaller [ y^+ ]. Kadri boiling onset inavyokaribia, energy-transfer activity inazidi kujilimbikiza closer to wall.

Directional heat-flux decomposition

Heat equation katika flux form imeandikwa:

\[ \rho c_p\frac{\partial T}{\partial t} = \nabla\cdot\left(\mathbf{j}_q+\mathbf{j}_{\mathrm{adv}}\right) \]

. Conductive na advective fluxes ni:

\[ \mathbf{j}_q=-k\nabla T \]

\[ \mathbf{j}_{\mathrm{adv}} =-\rho c_p\mathbf{u}(T-T_{\mathrm{in}}) \]

.

Kielelezo 3.13 kinapolinganisha absolute flux magnitudes, order ifuatayo spanning four orders of magnitude inapatikana:

\[ |j_{\mathrm{adv}}^x| > |j_{\mathrm{adv}}^z| > |j_{\mathrm{adv}}^y| \approx |j_q^y| > |j_q^z| > |j_q^x| \]

Streamwise convection

Hii ndiyo largest component kwa sababu mean flow velocity ni much larger than other velocity components. Inabeba joto downstream kutoka heater lakini haiondoi joto directly kutoka wall kwenda channel core.

Spanwise convection

Hii ndiyo second-largest component. Ina-support quasi-streamwise vortices kuhamisha temperature laterally kutoka low-speed streaks na kufanya temperature field more homogeneous across channel width.

Wall-normal convection na conduction

Components hizi mbili zina similar magnitude. Katika quasi-steady transition, wall-normal convection ni stronger relative to applied heat flux. Flow field ina enough time kuadjust to slow loading, hivyo joto linaondolewa more effectively kutoka hot wall.

Katika fast transition, relative development ya wall-normal convection inabaki more limited, huku conduction na local energy storage zikichukua bigger role. Hii inasaidia hot layer kubaki near wall despite higher wall superheat.

Conduction components

Spanwise na streamwise conduction ndiyo smallest flux components. Spanwise conduction inafikia maximum approximately [ y^+\approx7 ], na streamwise conduction approximately [ y^+\approx9 ].

Comparison ya fast na quasi-steady heating

Nguvu za utafiti

• Numerical model iliendeshwa directly na experimental wall heat-flux history.
• Wall temperatures zililinganishwa na experimental measurements.
• Temperature-dependent viscosity na thermal conductivity zilijumuishwa ili kuunda two-way thermal–hydrodynamic coupling.
• Si mean values pekee bali instantaneous three-dimensional temperature na vortex structures zilichunguzwa.
• Very high cell counts na fine near-wall grid zilitumika.
• Subgrid-model contributions zilitathminiwa quantitatively.
• Velocity, vorticity, shear stress, normal stresses, Stanton number na directional heat fluxes zilianaliziwa kwa namna ya complementarity.
• Two-parameter high-fit sigmoid relation ilipendekezwa kwa temperature profiles.
• Fast na quasi-steady transitions zilionyeshwa kuelekea boiling onset kupitia different single-phase pathways.

Mapungufu ya utafiti

• Study ni preprint ambayo haijapitia peer review.
• Simulations ni single-phase na hazifunikii kipindi baada ya first bubble kuunda.
• Bubble nucleation, bubble growth, surface cavities na two-phase interface dynamics hazijamodeliwa.
• Density na specific heat zimewekwa constant; only viscosity na thermal conductivity zinategemea temperature.
• Flow imechukuliwa incompressible.
• Gravity imepuuzwa kutokana na low bulk Richardson number.
• LES model inaweza kutorepresent fully local balance between turbulence production na dissipation katika rapid nonequilibrium transitions.
• Experimental heat-flux boundary condition ni spatial average; local losses at heater edges hazijajumuishwa.
• Results zinahusu only two channel geometries, specified Reynolds numbers, 10 bar pressure na 100 K inlet subcooling.
• Study mostly compares 5 na 100 ms extreme cases; detailed mechanism analysis ya 20 ms intermediate case ni more limited.
• Temporal averaging na Reynolds decomposition hazikutumika; focus ni spatial averages za instantaneous fields.
• Raw three-dimensional LES fields hazijawekwa kwenye open repository kutokana na large data volume.
• Generalizability ya sigmoid relation kwa different fluids, pressures na broader Reynolds-number range haijaonyeshwa.

Study inaonyesha nini?

• Rapid wall heating inabadilisha near-wall velocity field before boiling begins.
• Temperature-dependent viscosity reduction inahusiana na velocity increase na stronger spanwise vorticity.
• Couette-like constant-shear region inaweza kutokea katika buffer layer.
• Heated structures zinacompress wall-normal na kuenea spanwise.
• Katika fast transition, Stanton number inaongezeka approximately asilimia 30.
• Thermal entrance length ya fast transition ni shorter than quasi-steady transition.
• Temperature profiles katika thermally fully developed region zinaweza kuwakilishwa kwa high accuracy na two-parameter sigmoid law.
• Fast na quasi-steady transitions zinaunda different spatial superheating patterns.

Study haionyeshi nini?

• Haionyeshi exactly ni surface defect gani na microscopic mechanism gani bubble nucleation hutokea.
• Haihesabu jinsi heat-transfer coefficient inavyobadilika baada ya nucleate boiling kuanza.
• Haipredict critical heat flux au transition to film boiling.
• Haithibitishi sigmoid temperature law kuwa universal turbulence law.
• Haionyeshi kwamba all nuclear-reactor geometries zitaonyesha same asilimia 30 Stanton increase.
• Haijitenganishi conclusively kwa independent control simulations kwamba variable viscosity peke yake ndiyo cause ya all turbulence changes.
• Haijasimulate directly kwamba katika fast transition boiling definitely starts across whole surface at once; only single-phase temperature field inaonyesha initial condition toward that behavior.

Umuhimu wa kisayansi na kiteknolojia

Study inaonyesha kwamba average wall temperature peke yake haitoshi kwa safe management ya rapid heat loads. Two transitions zinazokaribia same wall temperature zinaweza kutofautiana katika near-wall turbulence na spatial structure ya superheated fluid.

Kwa nuclear-reactor safety, findings zinaonyesha kwamba models za predicting boiling onset wakati wa reactivity increase zinapaswa kuzingatia separately heating time scale, variable fluid properties na thermal entrance length.

Katika electronics cooling na rocket engines, rapid power changes zinaweza similarly kusababisha thermal storage karibu na wall na delayed flow reorganization. Hata hivyo, study haijamodel directly specific geometries za systems hizi; connection ni methodological.

Future work inahitaji kuunganisha single-phase LES results na bubble-nucleation models, surface microgeometry na interface-tracking methods ili kuchunguza jinsi calculated superheated film inavyobadilika kuwa actual boiling onset.

Mbinu na Matokeo ya Utafiti

Technical method summary

Main quantitative findings

Summary ya figure na table findings

Data processing na statistical evaluation

Study haitumii temporal averaging. Instantaneous fields zili-averagewa spatially katika spanwise direction na katika thermally fully developed downstream regions.

Relative wall-temperature difference ilitumika kwa experimental validation. [ R^2 ] coefficient of determination ilihesabiwa kwa sigmoid temperature profiles. Hakuna hypothesis test, p value, confidence interval au probabilistic uncertainty propagation iliyotumika.

Study inasema oscillations katika validation difference zinaweza kutokea kwa sababu largest turbulent motions hupita test section katika slower transitions. Oscillations hizi hazijatenganishwa quantitatively kwa separate spectral au statistical model.

Maelezo ya Chanzo na Mbinu

Jina kamili asilia la utafiti:Exponential Wall Heat-Flux Transients in Turbulent Channel Flow

Waandishi na mpangilio wao: Elie Roumet; Raksmy Nop; Nicolas Dorville; Christophe Bourcier; Elie Saikali; Marie-Christine Duluc.

Author–institution mappings:

• Elie Roumet: Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides; pia Conservatoire National des Arts et Métiers.
• Raksmy Nop: Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides.
• Nicolas Dorville: Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides.
• Christophe Bourcier: Université Paris-Saclay, CEA, Service de Génie Logiciel pour la Simulation.
• Elie Saikali: Université Paris-Saclay, CEA, Service de Génie Logiciel pour la Simulation.
• Marie-Christine Duluc: Conservatoire National des Arts et Métiers.

Institution addresses:

• Université Paris-Saclay, CEA, 91191 Gif-sur-Yvette, Ufaransa.
• Conservatoire National des Arts et Métiers, 75003 Paris, Ufaransa.

Corresponding author: Elie Roumet.

Contact address: elie.roumet@cea.fr

Equal first author: Hakuna equal-contribution au equal-first-authorship statement.

DOI:10.2139/ssrn.6947712

Official record:SSRN study page

Publication platform: SSRN.

SSRN publication date: 15 June 2026.

Publication year: 2026.

Page count: 34.

Source type: Research preprint in thermal hydraulics, computational fluid dynamics and turbulent heat transfer.

Journal: Hakuna peer-reviewed journal name, volume au issue information.

Original publisher: Hakuna separate peer-reviewed academic publisher iliyoverifywa.

Peer-review status: Study haijapitia peer review. Every page ya PDF ina warning “This preprint research paper has not been peer reviewed”.

Author contributions: Elie Roumet alifanya conceptualization, methodology, validation, formal analysis, investigation, visualization na initial draft writing; Raksmy Nop alifanya conceptualization, resources, funding acquisition, project administration na review; Nicolas Dorville alifanya conceptualization, project administration na review; Christophe Bourcier na Elie Saikali walifanya software, data curation na review; Marie-Christine Duluc alifanya conceptualization, supervision/advice na review.

Funding and computing resource: Study ilitumia TGCC high-performance computing resources chini ya allocations A0152A13031 na A0172A13031 zilizotolewa na GENCI.

Conflict of interest: Authors walitangaza hakuna known financial interests au personal relationships zinazoweza kuathiri study.

Data access: Raw three-dimensional LES fields hazijawekwa kwenye open repository kutokana na high data volume. Processed data supporting figures na main quantitative results zinapatikana from corresponding author upon reasonable request.

Artificial-intelligence use statement: Authors wameripoti kutumia OpenAI ChatGPT ili kuboresha readability ya manuscript; baadaye wali-review na ku-edit text na wakachukua full responsibility for content.

Makala hii ya Kituruki iliandaliwa kwa kuchunguza study nzima iliyopakiwa, pamoja na formulas, boundary conditions, numerical methods, graphs na field images, table values, appendices na author declarations. Scientific explanations ziliwekewa kikomo kwenye methods, data na interpretations zilizotolewa katika study; hakuna additional scientific finding kutoka external sources iliyoongezwa.

Main methodological limitation ya study ni kwamba only single-phase period before nucleate-boiling onset ndiyo imemodeliwa. Kwa hiyo results hazipaswi kutumika kama direct prediction ya bubble nucleation, two-phase heat transfer, critical heat flux au film boiling. Aidha, study bado haijapitia independent peer review.