Диссертация (1150484), страница 16
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4.1: Ôîðìà OR îáëàñòåé äëÿ ðàçëè÷íûõ ìîìåíòîâ âðåìåíè ïðè ïåðåñîåäèíåíèè â ñæèìàåìîé ïëàçìå ïðè ïðîèçâîëüíûõ íà÷àëüíûõ óñëîâèÿõ.125(1)×òîáû ïîëó÷èòü îðèãèíàë äëÿ Bz (t, x, z), íóæíî âîñïîëüçîâàòüñÿ (4.108):)∫ ((0)1BL̄b (s)Bz(1) (t, x, z) = ℜQ̄(s)F0 (t − τ (s))ds−ispπ Cv0L̄a (s) + L̄b (s))∫ (1L̄b (s)B (0)=ℜ−isQ̄(s)E ∗ (t − τ (s))dsπ Cv0L̄a (s) + L̄b (s)∫B (0) 1=ℜkern(Bz(1) Q̄(s)E ∗ (t − τ (s))dsv0 π C(4.121)ãäåkern(Bz(1) ) = isL̄b (s)L̄a (s) + L̄b (s)(4.122)kern - ìíîæèòåëü, êîòîðûé íóæíî ïîäñòâëÿòü â èíòåãðàë äëÿ ξz (4.108), äëÿ ïîëó÷åíèÿ âîçìóùåíèÿ ñîîòâåòñòâóþùèõ ÌÃÄ-ïàðàìåòðîâ.
Êðîìå òîãî â ïîäûíòåãðàëüíîì óðàâíåíèè íóæíî ïåðåéòè îò F0 ê E ∗ òàê êàê èç (4.106) ñëåäóåò, ÷òîpF0 (t − τ (s)) = E ∗ (t − τ (s)).Bx(1) (t, x, z) = B0∂ξz (t, x, z)∂z(4.123)Bx(1) (ω ′ , k, z) = B0 (−q)ξz (ω ′ , k, z)(4.124)L̄b (s)L̄a (s) + L̄b (s)(4.125)kern(Bx(1) ) = −q̄(s)Äàëåå îïðåäåëÿåì âîçìóùåíèå ïîòîêà ïëàçìû îñíîâûâàÿñü íà (4.34):vz(1) (t, x, z) =∂∂ξz (t, x, z) + v (0) ξz (t, x, z)∂t∂x(4.126)ãäå v0 - ñêîðîñòü òå÷åíèÿ â íà÷àëüíîé êîíôèãóðàöèè.vz(1) (ω ′ , k, z) = i(ω − kv)ξz (ω ′ , k, z) = −ip(i + sv)ξz (ω ′ , k, z)(4.127)L̄b (s)L̄b (s)= iω ′ (s)L̄a (s) + L̄b (s)L̄a (s) + L̄b (s)(4.128)kern(vz(1) ) = −i(i + sv)òàê êàê ω = −ip; ω ′ = −(i + sv).126(1)Äëÿ îïðåäåëåíèÿ vx , äîïîëíèòåëüíî íóæíî èñïîëüçîâàòü (4.43):vx(1) (t, x, z) =∂∂ξx (t, x, z) + v (0) ξx (t, x, z)∂t∂xvx(1) (ω ′ , k, z) = i(ω − kv)ξx (ω ′ , k, z)}{′22 2s−ω(s)v′Aξ(ω, k, z)= −pω ′ sc2sx222(c2s s2 − ω ′ (s))(c2s vA s2 − u2 ω ′ (s)){}2 2′2vA s − ω (s)L̄b (s)kern(vx(1) ) = −ω ′ sc2s(c2s s2 − ω ′ 2 (s))(c2s vA2 s2 − u2 ω ′ 2 (s)) L̄a (s) + L̄b (s)(4.129)(4.130)(4.131)Îïðåäåëÿòü âîçìóùåíèÿ ïëîòíîñòè ñëåäóåò èñõîäÿ èç (4.36):()∂∂ρ(1) (t, x, z) = −ρ(0)ξx (t, x, z) + ξz (t, x, z)∂x∂z)(∂ξzρ(1) (ω ′ , k, z) = −ρ(0) −ikξx +∂z{}ω ′ 2 (s)(0)= −ρ(−q)ξz (ω ′ , k, z)2′2ω (s) − cs k}1/2{2 2′2vA s − ω (s)2ξz (ω ′ , k, z)= −ρ(0) ω ′ (s)2222′222′2(cs s − ω (s))(cs vA s − u ω (s)){}2 2′2vs−ω(s)L̄b (s)2Akern(ρ(1) ) = −ω ′ (s)(c2s s2 − ω ′ 2 (s))(c2s vA2 s2 − u2 ω ′ 2 (s)) L̄a (s) + L̄b (s)(4.132)(4.133)(4.134)Âîçìóùåíèå ãàçîâîãî äàâëåíèÿ âûðàæàåì ÷åðåç âîçìóùåíèå ïëîòíîñòè (4.37):kern(p(1) ) = c2s kern(ρ(1) ),(4.135)è ïåðâûé ïîðÿäîê ïîëíîãî äàâëåíèÿ âû÷èñëÿåì èñïîëüçóÿ (4.38):(())∂∂∂P (1) (t, x, z) = ρ(0) −u2ξx (t, x, z) + ξz (t, x, z) + vA2 ξx (t, x, z)∂x∂z∂x))( (∂ξz2(1)′(0)2+ vA (−ik)ξxP (ω , k, z) = ρ(4.136)u −ikξx +∂z127{= ρ(0)′2−ω (s)uω ′ 2 (s)2+ c2s vA2 k 2− c2s k 2}(−q)ξz = −L¯a ξz (ω ′ , k, z)L̄a (s)L̄b (s)(4.137)L̄a (s) + L̄b (s)Íà ýòîì ïîñòðîåíèå ðåøåíèÿ çàäà÷è ïåðåñîåäèíåíèÿ çàêàí÷èâàåòñÿ è ìû ïåðåkern(P (1) ) = −õîäèì ê àíàëèçó ýíåðãåòèêè.4.2Ýíåðãåòèêà ïåðåñîåäèíåíèÿ â ñæèìàåìîé ïëàçìåÏåðâûì íàøèì øàãîì, êàê è ðàíåå áóäåò ðàñ÷åò ýíåðãåòè÷åñêèõ ïðåîáðàçîâàíèéâíóòðè îáëàñòè âûòåêàíèÿ.
 äàííîé ñèòóàöèè îí ñèëüíî îñëîæíÿåòñÿ, ïîñêîëüêó,òåïåðü OR - îáëàñòü ïðåäñòàâëÿåò ñîáîé íàáîð ñóùåñòâåííî ðàçëè÷íûõ îáëàñòåé,â êàæäîé èç êîòîðûõ ýíåðãèÿ ïðåîáðàçóåòñÿ ïî-ñâîåìó.Òàê êàê ÌÃÄ-ïàðàìåòðû âî âñåõ ÷àñòÿõ OR - îáëàñòè óæå îïðåäåëåíû ïðè ðåøåíèè çàäà÷è Ðèìàíà, äëÿ ðàñ÷åòà ýíåðãåòèêè îñòàåòñÿ îïðåäåëèòü òîëüêî îáúåìûâñåõ ÷àñòåé OR.Èñõîäÿ èç ôîðìóë (4.110) - (4.113), îïðåäåëÿþùèõ ôîðìû ôðîíòîâ, ðàññ÷èòûâàåì îáúåìû çàêëþ÷åííûå ìåæäó íèìè:c(F0 (wA ) − F0 (wSa ))Ba1{∫)) }((∫cxxVa1 =dx −dx =F0 t −F0 t −Ba1wwAaSORORc=(wA − waS )G(t)Ba1cfaS − fC = Φa2 (wSa ) − Φa2 (v2 ) =(F0 (waS ) − F0 (v2 ))Ba2{∫)) }((∫cxxVa2 =F0 t −dx −F0 t −dx =Ba2wvaSa2ORORc=(waS − va2 )G(t)Ba1fA − fSa = Φa1 (wA ) − Φa1 (wSa ) =128(4.138)(4.139)(4.140)(4.141)fC − fbS = Φa2 (va2 ) − Φb2 (wbS ) =cVb2 =Ba2ccF0 (va2 ) −F0 (wbS ))Ba2Bb2()()∫xcxF0 t −dx −F0 t −dx =va2Bb2 ORwbSOR{}cc=v2 −wbS G(t)Ba2Bb2(4.142)∫ãäå∫(4.143)tG(t) =(4.144)F0 (τ )dτ0Òåïåðü ìîæåì âû÷èñëèòü èçìåíåíèå ìàãíèòíîé, êèíåòè÷åñêîé è òåïëîâîé ýíåðãèéâ ðàñ÷èòàííûõ îáúåìàõ ïî ôîðìóëàì:(W̃Bi (t) =2Ba,b−8π8πB̃x 2i)V (ORi )ρṽi2 (ORi )W̃Ki (t) =V21(pi − pa,b )V (ORi )W̃T i (t) =γ−1(4.145)(4.146)(4.147)Äëÿ òîãî ÷òîáû îöåíèòü, êàê ïåðåðàñïåäåëÿþòñÿ ðàçëè÷íûå âèäû ýíåðãèé,ðàñ÷èòàåì ñóììàðíûå çíà÷åíèÿ ìàãíèòíîé, êèíåòè÷åñêîé è òåïëîâîé ýíåðãèé âOR - îáëàñòè ïðè ðàçëè÷íûõ çíà÷åíèÿõ íà÷àëüíûõ óñëîâèé.Âñå ÷èñëåííûå ðàñ÷åòû áóäåì ïðîâîäèòü ñ áåçðàçìåðíûìè âåëè÷èíàìè, íîðìèðîâàííûìè òàê æå, êàê ýòî áûëî ñäåëàíî â ðàçäåëå 2.2.Äëÿ ïðîâåäåíèÿ ñðàâíèòåëüíîãî àíàëèçà çíà÷åíèé ýíðåãèé óäîáíî ïîñòàâèòüñëåäóþùèå îãðàíè÷åíèÿ íà íà÷àëüíûå äàííûå:Ba2 + Bb2 = const = 1 + 1,(4.148)ρa + ρb = const = 1 + 1,(4.149)P = B 2 + p = const = 5.1.(4.150)129Ñíà÷àëà, ðàcñ÷èòûâàåì êàê áóäåò ïåðåðàñïðåäåëÿòñÿ ýíåðãèÿ ïðè óâåëè÷åíèè àñèììåòðèè ìàãíèòíîãî ïîëÿ, ïðè óñëîâèè ÷òî ïëîòíîñòè ïëàçìû, ïî ðàçíûå ñòîðîíûðàçðûâà, îñòàþòñÿ ðàâíûìè:|Ba |/|Bb | = [1, 1.2, ...2];(4.151)ρa = ρb = 1.BapaBbpbWBORWKORWTOR1.004.10-1.004.10-1.321.501.981.073.95-0.924.25-1.301.491.961.133.82-0.854.38-1.261.481.891.173.71-0.784.48-1.221.451.831.213.63-0.734.57-1.171.411.761.243.56-0.684.64-1.131.391.691.263.50-0.634.70-1.081.371.63Òàáëèöà 4.1: Çíà÷åíèÿ ìàãèòíîé, òåïëîâîé è êèíåòè÷åñêîé ýíåðãèé â îáëàñòè âûòåêàíèÿ ïðèðàçëè÷íûõ íà÷àëüíûõ óñëîâèÿõ.Äàëåå, ðàññìîòðèì êàê ìåíÿþòñÿ çíà÷åíèÿ ýíåðãèé ïðè óâåëè÷åíèè àñèììåòðèèêàê ìàãíèòíîãî ïîëÿ, òàê è ïëîòíîñòè ïëàçìû, ïðè÷åì â ïåðâîì ñëó÷àå âûáåðåììàãíèòíîå ïîëå è ïëîòíîñòü âîçðàñòàþùèìè ñ îäíîé ñòîðîíû ðàçðûâà, è óáûâàþùèìè ñ äðóãîé ñòîðîíû:|Ba |/|Bb | = [1, 1.2, ...2];ρa /ρb = [1, 1.2, ...2].(4.152)Âî-âòîðîì ñëó÷àå ðàññìîòðèì ñèòóàöèþ, êîãäà ïî îäíó ñòîðîíó ðàçðûâà ìàãíèòíîé ïîëå âîçðàñòàåò, à ïëîòíîñòü ïëàçìû, íàîáîðîò óìåíüøàåòñÿ, â òî âðåìÿ130BapaρaBbpbρbWBORWKORWTOR1.004.101.00-1.004.101.00-1.321.501.981.073.951.07-0.924.250.92-1.311.481.961.133.821.15-0.854.380.85-1.281.472.011.173.711.20-0.784.480.80-1.251.451.871.213.631.25-0.734.570.75-1.201.421.811.243.561.30-0.684.640.70-1.181.401.761.263.501.33-0.634.700.66-1.141.371.71Òàáëèöà 4.2: Çíà÷åíèÿ ìàãèòíîé, òåïëîâîé è êèíåòè÷åñêîé ýíåðãèé â îáëàñòè âûòåêàíèÿ ïðèðàçëè÷íûõ íà÷àëüíûõ óñëîâèÿõ.êàê ïî äðóãóþ ñòîðîíó ðàçðûâà, ïîëå óìåíüøàåòñÿ, à ïëîòíîñòü ïëàçìû âîçðàñòàåò:|Ba |/|Bb | = [1, 1.2, ...2];(4.153)ρa /ρb = [1, 0, 8, ...0, 5].BapaρaBbpbρbWBORWKORWTOR1.004.101.00-1.004.101.00-1.321.501.981.073.950.92-0.924.251.07-1.301.491.941.133.820.85-0.854.381.15-1.251.481.871.173.710.80-0.784.481.20-1.201.471.801.213.630.75-0.734.571.25-1.161.441.731.243.560.70-0.684.641.30-1.101.421.651.263.500.66-0.634.701.33-1.051.401.58Òàáëèöà 4.3: Çíà÷åíèÿ ìàãèòíîé, òåïëîâîé è êèíåòè÷åñêîé ýíåðãèé â îáëàñòè âûòåêàíèÿ ïðèðàçëè÷íûõ íà÷àëüíûõ óñëîâèÿõ.Ãëÿäÿ íà ïðâåäåííûå âûøå òàáëèöû ìîæíî âèäåòü, ÷òî óâåëè÷åíèå àñèììåòðèè íà÷àëüíûõ ïàðàìåòðîâ ïðèâîäèò ê óìåíüøåíþ âñåõ âèäèâ ýíåðãèè âíóòðè131OR-îáëàñòè.
Íàèáîëåå âûðàæåííîå âëèÿíèå íîñèò àñèììåòðèÿ ìàãíèòíîãî ïîëÿ,êîòðàÿ âëèÿåò íà çíà÷åíèÿ âñåõ ýíåðãèé, â òî âðåìÿ êàê àñèììåòðèÿ ïëîòíîñòèïëàçìû, ïðàêòè÷åñêè íå îêàçûâàåò âëèÿíèÿ íà çíà÷åíèå êèíåòè÷åñêîé ýíåðãèèóñêîðåííûõ ïëàçìåííûõ ïîòîêîâ.Ñëåäóþùèì øàãîì, ðàñ÷èòàåì êàê ïåðåðàñïðåäåëÿåòñÿ ýíåðãèÿ â îáëàñòè âòåêàíèÿ.Êàê óæå áûëî ïîêàçàíî (3.16), ïîëíîå èçìåíåíèå ìàãíèòíîé ýíåðãèè â îáëàñòèâòåêàíèÿ ìîæíî âûðàçèòü ÷åðåç z -êîìïîíåíòó âåêòîðà ñìåùåíèÿ:∆WBIRBa,b=−4π∫∞ξz (t, x, 0)dx.(4.154)0Åñëè æå ìû áóäåì ñ÷èòàòü èçëèøåê ìàãíèòíîé ýíåðãèè â ñòîëáå (x : [ x, x+dx]; z :[ 0, ∞)), òî ïîëó÷èì ïðîñòî:∆WBIRcol = −Ba,bξz (t, x, 0)dx.4π(4.155)Èçìåíåíèå òåïëîâîé ýíåðãèè â îáëàñòè âòåêàíèÿ â öåëîì è â ñòîëáå (x : [ x, x +dx]; z : [ 0, ∞)), òàê æå ðàñ÷èòûâàåòñÿ ïðè ïîìîùè z -êîìïîíåíòû âåêòîðà ñìåùåíèÿ (3.27):∆WTIRγp0=γ−1∆WTIRcol∫∞ξz (t, x, 0)dx,(4.156)−∞γp0=ξz (t, x, 0)dx.γ−1(4.157)Òåïåðü, çíàÿ óæå ðàñ÷èòàííóþ ðàíåå z -êîìïîíåíòó âåêòîðà ñìåùåíèÿ, ìîæåì èññëåäîâàòü, êàê ïåðåðàñïðåäåëþòñÿ ýíåðãèè â ïðîöåññå ïåðåñîåäèíåíèÿ â îáëàñòèâòåêàíèÿ.Ìîæíî âèäåòü êàê ìàãíèòíàÿ è òåïëîâàÿ ýíåðãèè, èõ ñóììàðíîå èçìåíåíèå â ñòîëáå, ðàñïðåäåëåíû îòíîñèòåëüíî OR - îáëàñòè.
Òàê æå, êàê è âî âñåõïðåäûäóùèõ ñëó÷àÿõ, ìû íàáëþäàåì îáëàñòü ïîâûøåííîé ýíåðãèè íàä è ïîä óäàðíûìè âîëíàìè, è óìåíüøåíèå ýíåðãèè â îáëàñòè ðàçëåòà.132Ïðîñóììèðîâàâ ïîëîæèòåëüíûå çíà÷åíèÿ ýíåðãèé â îáðàçîâàâøåéñÿ íàä OR îáëàñòüþ âîëíå ñæàòèÿ, ìîæåì îöåíèòü, ñêîëüêî ýíåðãèè ïåðåíîñèòñÿ âäîëü òîêîâîãî ñëîÿ âìåñòå ñ óäàðíîé âîëíîé, õîòÿ è âíå åå.
Áóäåì èñïîëüçîâàòü òå æåíà÷àëüíûå ïàðàìåòðû è îãðàíè÷åíèÿ íà íèõ, ÷òî è äëÿ ðàñ÷åòà ýíåðãèé âíóòðèOR - îáëàñòè (4.148) - (4.150).|Ba |/|Bb | = [1, 1.2, ...2];(4.158)ρa = ρb = 1.BapaBbpbIRWaBIRWbBIRWaTIRWbT1.004.10-1.004.100.950.954.864.861.073.95-0.924.251.000.874.305.491.133.82-0.854.381.040.793.876.061.173.71-0.784.481.050.723.526.551.213.63-0.734.571.040.643.216.961.243.56-0.684.641.020.582.947.291.263.50-0.634.700.990.512.727.57Òàáëèöà 4.4: Çíà÷åíèÿ ìàãèòíîé è òåïëîâîé ýíåðãèé â îáëàñòè âòåêàíèÿ ïðè ðàçëè÷íûõ íà÷àëüíûõ óñëîâèÿõ.Èç ïåðâîé ïðåäñòàâëåííîé òàáëèöû âèäíî, ÷òî óâåëè÷åíèå àñèììåòðèè ìàãíèòíîãî ïîëÿ ïðèâîäèò ê óìåíüøåíèþ ñóììàðíîé ìàãíèòíîé ýíåðãèè â âîëíå ñæàòèÿ, â òî âðåìÿ êàê êîëëè÷åñòâî òåïëîâîé ýíåðãèè óâåëè÷èâàåòñÿ.
Êðîìå òîãî,ìàãíèòíîé ýíåðãèè â âîëíå áîëüøå ñî ñòîðîíû áîëüøåãî ïîëÿ, à òåïëîâîè ýíåðãèèíàîáîðîò áîëüøå ñî ñòîðîíû ìåíüøåãî ïîëÿ, ïðè÷åì ïðèðîñò òåïëîâîé ýíåðãèè íàñòîëüêî âåëèê, ÷òî ñóììàðíî, òåïëîâîé è ìàãíèòíîé ýíåðãèè îêàçûâàåòñÿ áîëüøåñî ñòîðîíû ìåíüøåãî ïîëÿ.|Ba |/|Bb | = [1, 1.2, ...2];133ρa /ρb = [1, 1.2, ...2].(4.159)BapaρaBbpbρbIRWaBIRWbBIRWaTIRWbT1.004.101.00-1.004.101.000.950.954.864.861.073.951.07-0.924.250.920.960.924.135.801.133.821.15-0.854.380.850.950.883.586.681.173.711.20-0.784.480.800.940.823.157.491.213.631.25-0.734.570.750.910.762.818.231.243.561.30-0.684.640.700.880.702.538.901.263.501.33-0.634.700.660.840.652.319.51Òàáëèöà 4.5: Çíà÷åíèÿ ìàãèòíîé è òåïëîâîé ýíåðãèé â îáëàñòè âòåêàíèÿ ïðè ðàçëè÷íûõ íà÷àëüíûõ óñëîâèÿõ.|Ba |/|Bb | = [1, 1.2, ...2];(4.160)ρa /ρb = [1, 0, 8, ...0, 5].BapaρaBbpbρbIRWaBIRWbBIRWaTIRWbT1.004.101.00-1.004.101.000.950.954.864.861.073.950.92-0.924.251.071.050.834.525.251.133.820.85-0.854.381.151.130.734.215.591.173.710.80-0.784.481.201.160.643.895.861.213.630.75-0.734.571.251.160.563.596.071.243.560.70-0.684.641.301.160.493.346.221.263.500.66-0.634.701.331.160.433.156.34Òàáëèöà 4.6: Çíà÷åíèÿ ìàãèòíîé è òåïëîâîé ýíåðãèé â îáëàñòè âòåêàíèÿ ïðè ðàçëè÷íûõ íà÷àëüíûõ óñëîâèÿõ.Èç ñëåäóþùèõ äâóõ òàáëèö âèäíî, ÷òî íàëè÷èå àñèììåòðèè ïëîòíîñòè ïëàçìû,ïðèâîäèò íå ñòîëüêî ê îáùåìó óìåíüøåíèþ êîëëè÷åñòâà ýíåðãèè â âîëíå ñæàòèÿ, ñêîëüêî, ê åå ïåðåðàñïðåäåëåíèþ ìåæäó ìàãíèòíîé è òåïëîâîé.
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