I'll try to answer your question, sorry if it doesn't satisfy you. I'll primarily answer this from the Ottoman perspective

Generally, not much of a 'big' battle, but there are some raids and counter-raids by the Ottoman and Russian navy. See, for the Ottomans, the Black Sea is important because they relied on it for shipping coals. Ottoman coals were mostly mined in Northern Anatolia like Eregli, Kozlu, and most importantly Zonguldak. Because of the relatively poor infrastructure of the region, they must ship it via sea. The Russians noticed this and in 1915 put effort to blockade and raid Ottoman shipping. To combat this, the Ottomans employed as many warships as possible to protect the convoy, sometimes successful, but overall the Ottoman command avoided battle with the Russians as much as possible because of the lack of capable ships. In the war, the only ships that can face its Russian counterparts are the Yavuz, Midilli, Hamidiye, and Mecidiye. I'll try to break down events by year. Do note that I will only list ship engagements or major naval events so no detail on raids, convoy escorts, and land campaigns

1914

For the ottomans, naval action started before WWI, with the raids on Russian coasts on October 1914. On October, Yavuz, escorted by the destroyers Taşoz and Samsun fired upon shore batteries of Sevastopol, retreating after the batteries fired back and hit Yavuz three times. Afterwards, many different Ottoman warships scored raids to the Russians. The minelayer Nilüfer laid mines off Sevastopol, the cruiser Hamidiye bombarded Feodosia, the destroyers Muâvenet-i Millîye and Gayret-i Vatâniye shelled Odessa and sunk a gunboat, and Berk-i Satvet and Midilli shelled Novorossiysk. These raids were of meagre results, but it did its job of beginning hostilities with Russia and gave confidence to the Ottoman navy. After the war were declared, Ottoman warships were escorting convoys, with Midilli shelling Poti on 7 November. On 17 November, after learning that the Russian Black Sea Fleet was bombarding Trebizond, An Ottoman fleet consisted of Yavuz and Midilli were dispatched under Admiral Souchon, hoping to intercept the Russian fleet before it reached Sevastopol. On 18 November, Midilli sights enemy warships off Crimea, and Yavuz steamed to engage. The Russian flagship Estavfi opened fire first, and one of the shells struck Yavuz with light casualties. Afterwards Yavuz returned fire, a shell penetrated Evstafi’s funnel and bursted, sending shrapnel over the deck and knocking out its wireless antenna. Yavuz kept firing and Estavfi suffered a lot of damage, but Souchon withdrawn after five minutes because he feared that flames from Estafvi's explosion could affect the Yavuz. This action put Estafvi out of commission for some time. Afterwards aside from Hamidiye shelling Tuapse on 20 November, most of the Ottoman warships were protecting convoys

1915

Early 1915 were spent protecting convoys, with minor engagement on 6 January between Ottoman cruisers Midilli and Hamidiye and Russian cruiser Pamyat Merkuriya and Estafvi. Afterwards there are raids by both sides, with Russian fleet shelling Ottoman coastal ports and coal mines mainly to divert Ottoman attention from Gallipoli, and Ottoman fleet shelling Odessa and Constanza. The last action of a large Ottoman fleet was in 28 March where the cruisers Hamidiye and Mecidiye and the destroyers Muâvenet-i Millîye, Yâdigâr-ı Millet, Samsun, and Taşoz raided Odessa. It inflicted little casualties and Mecidiye sunk after hit by a mine (later it was recovered and refitted by the Russians). On 10 May, Yavuz alongside destroyer Nümûne-i Hamiyet took a mission to intercept Russian task force containing battleships Tri Sviatitelia and Panteleimon was approaching the Bosphorus. Nümûne-i Hamiyet sighted three battleships Evstafi, Joann Zlatoust, and Rostislav. Nümûne-i Hamiyet fired at them but soon realized that Tri Sviatitelia and Panteleimon had caught up, so they now faced five battleships. Knowing this, Yavuz fired at them which all missed, and after taking a heavy-caliber hit to the forecastle and another to the forward armored belt the two warships turned away from the Bosphorus, followed by their Russian counterparts. Both sides suffered minimal damages, and while Yavuz had performed poorly on the engagement, they successfully disrupted Russian operations by leading them into a goose chase. On June 10, Midilli intercepted the destroyers Derzkiy and Gnevniy off the Bosphorus. Midilli fired first, crippling Derzkiy and hitting Gnevniy’s steam pipes, causing it to lose power. However, Midilli's captain Leberecht von Klitzing didn't finish off Gnevniy, not wishing to put his vessel at any further risk and thus missing the chance to sunk one of Russian navy's best destroyer. Some more convoy escort and raids followed afterwards. On 5 September, the cruiser Hamidiye alongside destroyers Muâvenet-i Millîye and Nümûne-i Hamiyet spotted Russian destroyers Bystry and Pronzitelny while escorting convoys and attacked. However, luck is not with the Ottomans and Hamidiye's guns malfunctioned. The Russian destroyers then began a steady fire but sailed just beyond the range of Hamidiye’s secondary armament and the guns aboard the Ottoman destroyers. Afterwards the Russian submarine Nerpa was approaching and the Ottoman fleet retreated. The engagement was a failure for the Ottomans. The year was spent protecting convoys.

1916

By 1916, the Russians had gained upper hand on the Black Sea with the arrival of several new warships including the battleships Imperatritsa Mariya and Imperatritsa Ekaterina Velikaya. On 8 January, Yavuz followed Russian destroyers Pronzitelni and Leitenant Shestakov en route to raid Zonguldak. Yavuz gave chase, but later stumbled into Imperatritsa Ekaterina. Both sides opened fire, but after fifteen minutes Ekaterina’s rate of fire declined because of mechanical problems. No hits were scored on either side. But the encounter demonstrated that the Ottomans no longer have the edge with Yavuz, the new Russian battleships may have been slower, but they outgunned the most advanced ship in the Ottoman navy. For some time there were little naval combat, as the Russian navy were supporting campaigns in Lazistan on 17 to 20 January and aiding Russian amphibious landing on Trebizond on 18 April. At this time, Ottoman cruiser Midilli along with German U-boat U33 were transporting troops to Trebizond and picking off small Russian ships and convoys, sinking Russian minesweeper T233. On 19 April Midilli encountered Imperatritsa Mariya and barely escaped with high speed zig-zag steering. On 4 July Yavuz and Midilli shelled the town of Tuapse and sunk several Russian convoys. On 22 July while en route to mine Novorossiysk, the Midilli engaged the Russian destroyer Schastliviy. However, the Schastliviy contacted Imperatritsa Mariya, resulting in Midilli getting shelled by Mariya. Only by laying smokescreen and mines is Midilli able to escape. In order to prevent further Ottoman offensive action, from 31 July to 8 August the Russians laid over 1,000 mines in a semicircle north of the Bosphorus. The Russian minefields brought all waterborne coal and food supplies to Istanbul to a halt. In order to combat the mines, many torpedo boats and smaller vessels are refitted as minesweepers. From August to November, five ships including two minesweepers were sunk by mines in and around the Bosphorus. During this lull, the action in the Black Sea were dominated by U-boat raids. For example on 31 August UB45 sank the Russian transport Tevere, and later Gioconda on 2 September, both full of men going to the Caucasus. Later on a three day storm from 23 to 25 September detonated several hundred mines, making limited shipping possible. After further Ottoman efforts, the mines were cleared by October. The year ended with Russian raid into Terme in 2 November, the closest the Russian ground forces would come to Istanbul and the sinking of two Ottoman gunboats on 21 December by Russian destroyer Pamyat Merkuriya

(1/2)

(2/2)

1917

The year marked with aggressive Russian naval actions and it subsequent defeat after the armistice. Russian submarines stalked the Bosphorus from January to February, sinking many merchant ships and laid additional mines, again blocking Ottoman shipping. In early March, The Russians shelled Giresun and Tirebolu. Additionally, the Russians intended to attack the Bosphorus on 26 March, but uncertainty over naval superiority placed the operation on hold. During April and May, Ottoman minesweepers swept clear the minefields and channels over the Bosphorus. The Russians responded by sending a fleet of small motorboats to re-mine the approaches. The operation was conducted at night without being seen by coastal patrols and the re-mining remained unknown to the Ottomans until 27 May, when a mine self-detonated. On 17 May Germany dispatched six minesweepers to Istanbul by rail. This group, combined with Ottoman minesweepers, managed to clear the mines in two weeks. The Russians then tried again to re-mine the Bosphorus but Ottoman aerial attacks curtailed further mining efforts. The growing internal chaos in Russia in mid-1917 brought a respite to the hard-pressed Ottoman navy, especially the minesweepers. On 23 June, the cruiser Midilli was sent out to capture and destroy the Russian radio station on Snake Island and then lay a minefield between the island and the mouth of the Danube. After successfully doing its mission, Midilli makes home On 25 June, but on the way encountered Imperatritsa Ekaterina (now Svobodnaya Rossiya) and the destroyer Gnevnyi, en route to the Bosphorus to cover a minelaying mission. Imperatritsa Ekaterina fired upon Midilli but the cruiser managed to stay just out of range with its superior speed. As Midilli approached the Bosphorus, the destroyer Basra sailed out to support Midilli. Unfortunately Basra could only make 18 knots and Midilli had to slow down. The slower speed allowed Imperatritsa Ekaterina to catch up and Midilli was put in the awkward position of defending Basra, which was supposed to be escorting Midilli. Midilli then released a smokescreen to cover them and after a harried pursuit, Midilli and Basra reached the entrance to the Bosphorus and the Russians turned away. Russian raid continues, but little action was taken with coal shortages for the Ottoman navy. On 29 October fearing the route to Constanza was mined, the Ottomans deployed the gunboat Hamidabad, towing three small German-built minesweepers to the base at Igneada. On 31 October, the Russian destroyers Pylkiy and Bystriy intercepted the Hamidabad, and the gunboat was set ablaze. The minesweepers escaped when the craft were released to flee. The Midilli was then deployed to hunt the Russians, but the destroyers fled north to safety. Afterwards, with the Russian Revolution reaching Crimea, the Black Sea Fleet was finished. Ottoman shipping then continued in relative safety and delivered much-needed coal to Istanbul. On 16 December 1917, an armistice was concluded. By the end of the war in the Black Sea, the Ottoman Empire had lost most of its merchant fleet and the navy was battered beyond recognition, but still standing. The year was spent repairing and refitting remaining warships. The morale of the Ottoman crew remained high, the three years of action had raised public opinion of the navy, and the level of training had done much to improve the fighting spirit of Ottoman seamen. The Ottoman navy still saw action in 1918, but much of it was in the Dardanelles.

On the Russians and Gallipoli, yes, there were plans for it and Churchill requested Russian assistance, but the Russian command believed that the Russian navy was not up to the task and viewed attack on the Bosphorus to be very hard or impossible. On 1915 Grand Duke Nicholas emphasized the difficulties facing the Black Sea Fleet: the Russian dreadnoughts were not finished, and there were no modern submarines and only an insufficient number of fast modern destroyers. The Black Sea Fleet was no more than the equal of the Ottoman fleet and that’s only when all ships were together. Russian ships could only carry coal for four days at sea, coaling at sea was impossible in bad weather, their nearest base was more than twenty-four hours from the entrance to the Bosphorus. Additionally, Ottoman coastal batteries were viewed as superior to Russian naval guns at the time. Moreover, they considered a minimum of two army corps necessary for a successful landing, and these troops could not be withdrawn from the principal theater of the war. Otherwise, the Russians claimed to fully support the Allied expedition. On 28 February Grand Duke Nicholas promised the British the Black Sea Fleet would attack Istanbul and an expeditionary force would be landed but that this would not take place until the British and French had broken through the Dardanelles and their fleets were before the Princess Islands in the Sea of Marmara. With the Anglo-French forces bogged down at the Dardanelles, on 25 April the British requested that the Russians apply pressure upon the Bosphorus to check the flow of reinforcements to Gallipoli. The Russian fleet then shelled the forts at the entrance of the Bosphorus as well as at Kava Narrows. Additionally, they also shelled the right flank of the Çatalca line and adjacent coastal defensive positions. But shore bombardment without follow-on forces become little more than a naval menace, and because the Russians are still facing the Austro-German assault on the Eastern front, no men will be sent to the Bosphorus. Additionally, the Russian fleet also had serious coal limitations that prevented any long-term operations off the Bosphorus. So overall, the Russian operations on the Bosphorus during the Gallipoli campaign only caused little damage, though on the other hand Russian raids and shelling on Eastern Anatolia brought devastating effects to the Ottoman Empire

Sources:

The Ottoman Steam Navy, 1828-1923 by Bernd Langensiepen and Ahmet Guleryuz

Ottoman Navy Warships 1914–18 by Ryan K. Noppen

A Naval History of World War I by Paul G. Halpern